Socrates owns just one ship. The ship is worth $200 million dollars. If the ship sinks,
Socrates loses $200 million. The probability that it will sink is .02. Socrates' total wealth, including the value of the ship is $225 million. He is an expected utility maximizer with utility U(W) equal to the square root of W. What is the maximum amount that Socrates would be willing to pay in order to be fully insured against the risk of losing his ship?
see my earlier post to this very question.
GG GL
To determine the maximum amount that Socrates would be willing to pay in order to be fully insured against the risk of losing his ship, we need to calculate his expected utility with and without insurance.
Without Insurance:
If Socrates doesn't take out insurance and the ship sinks, he would lose $200 million, reducing his wealth to $25 million (225 million - 200 million). If the ship doesn't sink, his wealth remains at $225 million. Thus, the expected utility without insurance (EUWI) can be calculated as follows:
EUWI = P(sink) * U(W - loss) + P(no sink) * U(W)
= 0.02 * U(25 million) + 0.98 * U(225 million)
With Insurance:
If Socrates takes out insurance and pays a premium (P) to be fully insured against the risk of losing his ship, he would effectively transfer the risk to the insurance company. In case of sinking, Socrates would receive the insured amount (I) of $200 million, and his wealth would still be reduced to $25 million.
Thus, after taking out insurance, Socrates' expected wealth would be (W - P + I) in case of sinking, and W - P in case of no sinking. The expected utility with insurance (EUWI) can be calculated as follows:
EUWI = P(sink) * U(W - P + I) + P(no sink) * U(W - P)
To find the maximum amount Socrates is willing to pay to fully insure against the risk of losing his ship, we need to set the expected utility without insurance equal to the expected utility with insurance:
EUWI = EUWI
0.02 * U(25 million) + 0.98 * U(225 million) = 0.02 * U(25 million) + 0.98 * U(225 million - P)
Simplifying the equation, we can solve for P:
U(225 million - P) = U(225 million)
Square both sides:
(225 million - P) = 225 million
P = 0
From the equation, we can see that Socrates would not be willing to pay anything to be fully insured against the risk of losing his ship since his expected utility remains the same with or without insurance. This implies that Socrates is risk-neutral when it comes to the loss of his ship.