Insurance and pooling of risks

Suppose an insurance company is considering offering a new disability insurance policy which pays out if the policy holder is injured and unable to work. It knows that there are 200 potential buyers, and that the buyers fall into two categories: high risk and low risk. The probability that a high risk individual will be injured is 15% and the probability that a low risk individual will be injured is 5%. The company also know that there are 100 people of each type but it can’t tell which type someone is when they apply for insurance.

1. Suppose the company offered a policy giving $30,000 of coverage and all 200 potential buyers were required to buy it (possibly as part of a group insurance plan at their employer). If the rate charged for each dollar of coverage is r (eg, the premium on the $30,000 policy would be 30000*r), derive an equation showing the total expected value of the policy to the insurance company. At what value of r would the policy be actuarially fair from the point of view of the insurance company?
2. Now suppose the insurance is optional but the rate per dollar of coverage is set at the value from part (1). In addition, assume that potential buyers are risk neutral and know their own type (eg, a high-risk person is aware that she is high-risk). Draw the decision tree for a low risk individual and determine whether or not she will buy the insurance policy. Now determine whether the high risk individuals will buy the policy. Be sure to show all your work. Discuss the implications of your results.

Solution