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?