Question 5:
Solve Problems 4.93 and 4.94, given below.
Insurance. The business of selling insurance is based on probability and the law of large numbers. Consumers buy insurance because we all face risks that are unlikely but carry high cost. Think of a fire destroying your home. So we form a group to share the risk: we all pay a small amount, and the insurance policy pays a large amount to those few of us whose homes burn down. The insurance company sells many policies, so it can rely on the law of large numbers. Exercises 4.91 to 4.94 explore aspects of insurance.
4.93 Life insurance. Assume that a 25-year-old man has these probabilities of dying during the next five years:
Age at death | 25 | 26 | 27 | 28 | 29
Probability | 0.00039 | 0.00044 | 0.00051 | 0.00057 | 0.00060
a) What is the probability that the man does not die in the next five years?
b) An online insurance site offers a term insurance policy that will pay $100,000 if a 25-year-old man dies within the next five years. The cost is $175 per year. So the insurance company will take in $875 from this policy if the man does not die within five years. If he does die, the company must pay $100,000. Its loss depends on how many premiums the man paid, as follows:
Age at death | 25 | 26 | 27 | 28 | 29
Loss | $99,825 | $99,650 | $99,475 | $99,300 | $99,125
What is the insurance company's mean cash intake from such polices?
4.94 Risk for one versus thousands of life insurance policies. It would be quite risky for you to insure the life of a 25-year-old friend under the terms of Exercise 4.93. There is a high probability that your friend would live and you would gain $875 in premiums. But if he were to die, you would lose almost $100,000. Explain carefully why selling insurance is not risky for an insurance company that insures many thousands of 25-year-old men.