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Math Help - Utility functions and insurance

  1. #1
    Junior Member
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    Dec 2008
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    Utility functions and insurance

    Hi, I think I've done this right, however, i need to know if I have as there is no point me thinking I am right if I'm not, obviously.

    Dealer:
    U(x) = 2ln(x)
    Initial Wealth = 500,000
    Books = 10,000
    Total Wealth = 510,000
    Pr(complete loss of books) = 0.02
    Pr(no loss) = 0.98

    Insurer:

    U(x) = ln(x + 200,000)
    Assets = 2,000,000


    1 - The insurer initially offers the dealer a quote of 250. Find out if the dealer would accept this offer.

    2 - Find the maximum offer the dealer would accept.

    3 - The dealer offers to pay 201, find out if the insurer would accept this lower price.

    1)

    P:E[U(a-x)]=U(a-P)

    E[U(a-x)] = (2ln(510000)*0.98)+(2ln(500000)*0.02)
    E[U(a-x)] = 26.2835

    26.2835 = 2ln(510000-P)
    exp(13.14177) = 510000 - P
    P = 201.948

    As the initial offer is greater than P the initial offer would be rejected.

    2)

    The maximum quote the dealer would accept is 201.94 (rounding down as up would be more than the maximum)

    3)

    Insurer without insuring
    U(x) = ln(2,000,000 + 200,000)
    U(x) = ln(2,200,000)
    U(x) = 14.60396792

    Insurer insuring
    U(x) = (ln(2,200,000 + 201) * 0.98) + (ln(2,200,000 + 201 - 10,000) * 0.02)
    U(x) = 14.60396817

    Utility function with insuring is greater than not insuring so the insurance company would accept the dealers quote of 201.




    Does this method seem to be the right one? Are my conclusions correct?
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  2. #2
    Grand Panjandrum
    Joined
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    Quote Originally Posted by Rapid_W View Post
    Hi, I think I've done this right, however, i need to know if I have as there is no point me thinking I am right if I'm not, obviously.

    Dealer:
    U(x) = 2ln(x)
    Initial Wealth = 500,000
    Books = 10,000
    Total Wealth = 510,000
    Pr(complete loss of books) = 0.02
    Pr(no loss) = 0.98

    Insurer:
    U(x) = ln(x + 200,000)
    Assets = 2,000,000


    1 - The insurer initially offers the dealer a quote of 250. Find out if the dealer would accept this offer.

    2 - Find the maximum offer the dealer would accept.

    3 - The dealer offers to pay 201, find out if the insurer would accept this lower price.

    1)

    P:E[U(a-x)]=U(a-P)

    E[U(a-x)] = (2ln(510000)*0.98)+(2ln(500000)*0.02)
    E[U(a-x)] = 26.2835

    26.2835 = 2ln(510000-P)
    exp(13.14177) = 510000 - P
    P = 201.948

    As the initial offer is greater than P the initial offer would be rejected.

    2)

    The maximum quote the dealer would accept is 201.94 (rounding down as up would be more than the maximum)

    3)

    Insurer without insuring
    U(x) = ln(2,000,000 + 200,000)
    U(x) = ln(2,200,000)
    U(x) = 14.60396792

    Insurer insuring
    U(x) = (ln(2,200,000 + 201) * 0.98) + (ln(2,200,000 + 201 - 10,000) * 0.02)
    U(x) = 14.60396817

    Utility function with insuring is greater than not insuring so the insurance company would accept the dealers quote of 201.




    Does this method seem to be the right one? Are my conclusions correct?
    You leave too much for your reader to deduce, try to explain what you are doing or trying to do at each major stage of your solution. (In particlar don't use the same symbol for the dealers and insurers utility functions).

    CB
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