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Fair Insurance Premium Under Different Circumstances

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Suppose your company is considering three health insurance policies. The first policy requires no tests and covers all preexisting illnesses. The second policy requires that all covered employees test negative for the HIV virus. The third policy does not cover HIV or AIDS related illnesses. All insurance policies are priced at their actuarially "fair" value. All individuals are slightly risk averse. An individual with the HIV virus requires, on average, $100,000 worth of medical care each year. An individual without the virus requires, on average, $500 worth of medical care each year.

a. Suppose that the incidence of HIV in the population is 0.005. Calculate the annual premium of the first policy. (hint: adverse selection)
b. If you don not have insurance that covers HIV related illnesses, the probability of getting HIV is 1%. If you have insurance that covers HIV related illness, suppose that the probability of getting HIV is 2%. Calculate the premium of the second policy. Show you calculations. (hint: moral hazard)
c. In question (b), suppose the insurance company wants to encourage low-risk behavior by individuals who have insurance. On average, it "costs" individuals $100 to engage in low-risk behavior. Assume that if people get HIV, they pay the deductible; and if they do not get HIV, they do not pay the deductible. How high must the deductible be to encourage low-risk behavior?
d. Calculate the premium of the third policy. Show your calculations.

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Solution Preview

a. In the first case the insurance policy will be just enough to cover the cost of medicines for the population. Given that 0.005 is the incidence rate of HIV in the population, we can say that 99.5% of the people do not have it and they will spend $500 on medicines. The 0.5% will spend $100000. Thus average spending is
100000*0.005 + 500 * 0.995 = $997.5
This will be premium in the first case.

b. The second policy ...