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# Probability utility function and decision trees

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Pierre has a utility function for total asset position of u(x)=ln(x), His assets currently consist of \$50,000 in cash and a rare violin he inherited from a rich uncle which is valued at \$100,000. He is debating whether to buy insurance for the violin at an annual premium, Pr. There is a 1% chance that his violin will be lost, damaged or stolen during a given year.

i) Find the annual premium p at which he will be indifferent between buying the insurance or not.

ii) The insurance company is offering a new scheme called probabilistic insurance. Under the terms of this scheme, Pierre will pay a premium of Pr/2where Pr is the amount you found in part i). In the event of a claim, the insurance company will toss a fair coin. If the coin lands showing a head, Pierre will pay the other half of the premium and be insured for the claim. If the coin lands showing a tail, his Pr/2 payment will be returned and he will not be insured for the claim. Draw a decision tree for this problem using the Pr value found in part i). Should Pierre buy the probabilistic insurance?

In the computations of utility for this problem you will need to retain at least six decimal places.

https://brainmass.com/statistics/random-variables/probability-utility-function-and-decision-trees-563044

#### Solution Preview

Pierre has a utility function for total asset position of u(x)=ln(x), His assets currently consist of \$50,000 in cash and a rare violin he inherited from a rich uncle which is valued at \$100,000. He is debating whether to buy insurance for the violin at an annual premium Pr. There is a 1% chance that his violin will be lost, damaged or stolen during a given year.

i) Find the annual premium p at which he will be indifferent between buying the insurance or not.