Question

1. Suppose that an individual has the following utility function, ? = ?^0.5?.?, where U stands for utility and W for Wealth. The individual currently has a net wealth of $400,000. The individual believes there is a 5% chance they will get into a car accident this year. It is expected that a car accident would cost them $100,000 (dropping their overall wealth to $300,000).

a) How much would it cost to purchase an actuarially fair insurance policy to cover all losses from a car accident? (In other words, what are the expected losses)

b) How much would the individual be willing to pay for this policy? (What are the maximum loading fees that an insurance company could charge for this policy)

c) Provide an example utility function for a risk-averse, a risk-neutral, and risk-loving individual. You may use a graph or mathematical formula.

Answer 1

A) premium for actuarially fair insurance

= LOSS * loss probability

= 100,000* .05

= $ 5,000

B) maximum WTP = intital wealth - CE( certainty equivalent)

As EU = .05*√300,000 + .95*√400,000

= 198.66

So for CE, √CE = EU = 198.66

Thus CE = 39,465.7956

So max WTP = 40,000 - 39,465.7956

= 534.2044

C) for risk averse , utility function is concave in shape

So U(W) = W^{​​​​​​1/3}​​​​​​

For risk neutral, utility function is linear in wealth

U(W) = W

For risk lover, utility function is convex

U(W) = W^{​​​​​​2}