(Insurance) Let X = R+. Consider a house owner whose house has a risk of
burning down with probability 0.001. If the house burns down it is worth
$0 otherwise it is worth $1 million. The owner of the house is an expected utility maximizer with a vNM utility function u(x) = x^(1/2) over final wealth. Suppose now that there is an insurance company who may sell full insurance to this agent at a premium p. The contract is of the form: If the agent pays $p million then the
insurance company will pay the agent $1 million in case his house burns down. The insurance company does not pay back the p later, whether the house burns down or not.
(a) How do you describe the risk attitudes of the house owner?
(b) For which values of p, would the owner of the house be willing to buy the insurance?
(c) If the insurance company maximizes expected profits and knows (b), what
premium p* would it set? What are the company’s expected profits?
(a) To describe the risk attitudes of the house owner, we can analyze their utility function and the expected value of wealth under different scenarios.
Given that the utility function of the house owner is u(x) = x^(1/2), we can see that the function exhibits diminishing marginal utility. This implies that the house owner is risk-averse as the utility gained from wealth decreases as wealth increases.
Furthermore, the house owner faces the risk of their house burning down with a probability of 0.001. Since the utility function represents the owner's preferences over final wealth, the expected utility is computed by weighing the utility of each outcome with its associated probability. In this case, the expected utility can be calculated as follows:
E(u(x)) = (1 - 0.001) * u($1 million) + 0.001 * u($0)
Substituting the utility function, we have:
E(u(x)) = (1 - 0.001) * (1 million)^(1/2) + 0.001 * (0)^(1/2)
= 0.999 * 1000 + 0
= 999
The expected utility of the house owner is 999.
(b) To determine for which values of p the owner of the house would be willing to buy insurance, we compare the expected utility with and without insurance.
Without insurance, the expected utility is 999 as calculated in part (a).
With insurance, the owner would pay a premium p million and receive $1 million in case the house burns down. In this scenario, the final wealth would either be $1 million or $1 million - p, depending on the outcome.
The expected utility with insurance can be computed as follows:
E(u(x|insurance)) = (1 - 0.001) * u($1 million) + 0.001 * u($1 million - p)
Substituting the utility function, we have:
E(u(x|insurance)) = (1 - 0.001) * (1 million)^(1/2) + 0.001 * (1 million - p)^(1/2)
To determine the values of p for which the owner would be willing to buy insurance, we compare the expected utility without insurance (999) with the expected utility with insurance. If the expected utility with insurance is higher than 999, the owner would be willing to buy insurance.
(c) To find the premium p* that the insurance company would set in order to maximize expected profits, we consider the following:
The insurance company's profits are calculated as the difference between the premium received and the expected payout. In this case, the insurance company does not pay back the premium p, regardless of whether the house burns down or not.
The expected payout is given by:
Expected Payout = 0.001 * $1 million
Therefore, the insurance company's expected profits can be calculated as:
Expected Profits = Premium - Expected Payout
The insurance company would set the premium p* such that the expected profits are maximized. This would involve analyzing the relationship between the premium and the expected payout to determine the optimal premium that maximizes profits.
Note: To obtain the specific value of p* and the expected profits, the exact premium amount would need to be specified or further data provided.