Yuri owns just one ship. The ship is worth $200 million dollars. If the ship sinks, Yuri loses $200 million. The probability that it will sink is .02. Yuri's total wealth, including the value of the ship is $225 million. He is an expected utility maximizer with von Neuman Morgensten utility U(W) equal to the square root of W. What is the maximum amount that Yuri would be willing to pay in order to be fully insured against the risk of losing his ship

expected utility = (probability ship doesn't sink x √utility of the ship) + (probability ship sinks x √utility of ship sinking) = (98% x √$200,000,000) + (2% x √$0) = $13,859.29 13,939.82

fair premium of insurance policy = probability of loss x size of loss = 2% x $200,000,000 = $4,000,000

maximum premium = maximum utility - W* = $200,000,000 - W*

to find W*:
expected utility = √W
$13,859.29 = √W
W = $13,859.29²
W = $192,080,000

maximum premium = $200,000,000 - $192,080,000 = $7,920,000

maximum willingness to pay = (fair premium + maximum premium) / 2 = ($4 million + $7.92 million) / 2 = $5.96 million