Question

Herr Fenderbender considers himself a great driver. He has just received a Mercedes sports car as a gift from his uncle. His car is worth $90 000, but will have a scrap value of $10 000 in the event of a collision, an event that will occur with a probability of 0.4. He is offered the following insurance policy: In the event of a collision the insurance company will pay Herr Fenderbender $80 000. The insurance company is asking a premium of R = $27 500 which Fenderbender has to pay if the policy is accepted, whether or not there is a collision. Herr Fenderbender has a (von-Neumann Morgenstern) utility function of wealth given by: U = W ^ 1/2

1.Should Herr Fenderbender purchase this insurance? Explain carefully.

2.Should the insurance company be offering him this insurance (the insurance company has a lot of customers and hence is risk neutral; i.e. it has utility function U= x.

3.Repeat 1 and 2 for the case where the insurance premium is R = $37 100.

4.Find the range of values of the insurance premium, R, which will be accepted by Fenderbender AND which the insurance company would be willing to offer.

5.Suppose now that the insurance company knows that Herr Fenderbender is a good driver but believes that due to the moral hazard problem, after the insurance policy has been purchased the probability that there will be a collision will rise to 0.5. What does the insurance company believe its expected profit from sale of the policy is, and should the insurance company sell Fenderbender that policy? (R = $37 100)

Answer 1

1. Herr Fenderbender will purchase the insurance becaue at the time of collision, event that will occur with a probabitlity of 0.4 but there may be a chance to increase that. So it is wise to prefer the insurance in this amount.

2. Yes, the insurance company be offering him this insurance because the insurance company has a lot of customers and hence is risk neutral; i.e. it has utility function U= x.Von Neumann–Morgenstern utility function, an extension of the theory of consumer preferences that incorporates a theory of behaviour toward risk variance. n decision theory, the von Neumann–Morgenstern (or VNM) utility theorem shows that, under certain axioms of rational behavior, a decision-maker faced with risky (probabilistic) outcomes of different choices will behave as if he or she is maximizing the expected value of some function defined over the potential outcomes at some specified point in the future.

3. where the insurance premium is R = $37 100 the probability rate will vary and the premium is high and not wise to prefer the insurance amount.

4. the insurance company would be willing to offer to Fenderbender is R=$27 500, because when it become hight insurance premium probability that there will be a collision will rise to 0.5.

5. The insurance company do not sell Fenderbender the policy of R = $37 100 because the premium probability that there will be a collision will rise to 0.5.