Problem
Jack recently purchased a newly released vehicle in the market, TooFast, for $40,000. TooFast launched this year and there are 200,000 TooFast cars in the market, of which 80% have vehicle damage insurance purchased by the owner. Half of these car owners have a $200 deductible on their policy, 20% of these owners have a $1,000 deductible on policy, while the remainder do not have any deductible. The general rule of thumb is for every $100 deductible on your policy, your premium paid reduces by 3%.
In any given year, there are expected to be 10,000 TooFast cars damaged in accidents, with an average repair cost of $15,000. In terms of insured cars damaged, usually 15% are those with a $1,000 deductible, 25% are those with a $200 deductible, and 60% are those with no deductible.
Insurance
Company A is the only company providing insurance coverage for the TooFast cars in the market. Company A has an employee base assisting solely with TooFast car policies and claims, and are paid total salaries of $200,000 per year. Company A charges premiums such that all claims and employee salaries are fully paid for, plus a 10% profit margin for Company A.
Task
Assume TooFast cars come with a very expensive sound system (costs $4,000) which is not covered by insurance from Company A. This sound system can malfunction, and the dealership provides a separate insurance that covers repair costs.
There are two types of malfunctions: Malfunction A is full damage, and you get a new sound system, or Malfunction B which costs $750 to repair. You can choose from two deductibles - either none or $250. If you choose no deductible, you will be classified as a "regular" client (i.e., experience for losses are similar to customers that do not have insurance). Those with a deductible are viewed as "low-risk" clients.
For those with no deductible, the probability of Malfunction A and B in any given year is 8% and 27%, respectively. The annual insurance premium the car dealership is charging is $400 for such policies. Assume for simplicity that the premium that the dealership will charge is based on the expected claims they have to pay (i.e., no profits made by dealership).
For those with the deductible, the probability of Malfunction A and B in any given year is 4% and 24%, respectively. The annual insurance premium the car dealership is charging is $335 for such policies.
Assume Jack's net worth is $40,000 today and has a risk aversion parameter of 1.00. Should Jack purchase insurance, and which policy should Jack choose? What is the reservation price at which Jack would prefer to remain uninsured?