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’t 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 your calculations.
(Hint: Moral hazard.)
264 SECTION V UNCERTAINTY
c. In Question 20-3b, 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.
need help with this
a. To calculate the annual premium of the first policy, we need to take into account the adverse selection problem. Adverse selection occurs when individuals with higher risks are more likely to purchase insurance. In this case, the first policy covers all preexisting illnesses, including HIV. Given that the incidence of HIV in the population is 0.005 (or 0.5%), the probability of an individual having HIV is higher than the average population. Therefore, to calculate the annual premium, we need to adjust for this higher risk.
We can use the formula:
Premium = (Expected Cost of Medical Care for Individuals with HIV) / (1 - Probability of Having HIV)
The expected cost of medical care for an individual with HIV is $100,000.
So, the annual premium for the first policy would be:
Premium = $100,000 / (1 - 0.005) = $100,000 / 0.995 = $100,502.51 (approximately)
b. In this case, we need to consider the moral hazard issue. Moral hazard occurs when individuals are more likely to engage in risky behaviors when they have insurance coverage. If the probability of getting HIV without insurance is 1% and the probability of getting HIV with insurance is 2%, we can calculate the premium of the second policy.
Let's denote the premium as P, the expected cost of medical care without HIV as C1 (which is $500), and the expected cost of medical care with HIV as C2 (which is $100,000). The probability of getting HIV without insurance is 1% (or 0.01) and with insurance is 2% (or 0.02).
The expected cost of medical care with insurance is:
Expected Cost with Insurance = (C1 * (1 - Probability of Getting HIV)) + (C2 * Probability of Getting HIV)
Expected Cost with Insurance = ($500 * (1 - 0.02)) + ($100,000 * 0.02)
Expected Cost with Insurance = $10 + $2,000 = $2,010
To calculate the premium (P), we use the formula:
P = Expected Cost with Insurance / (1 - Probability of Getting HIV)
P = $2,010 / (1 - 0.02) = $2,010 / 0.98 = $2,051.02 (approximately)
Therefore, the premium of the second policy would be approximately $2,051.02.
c. To encourage low-risk behavior, the insurance company can introduce a deductible. A deductible is the amount that individuals have to pay out of pocket before the insurance coverage kicks in. The deductible helps discourage risky behavior as individuals bear some of the financial burden if they engage in high-risk activities.
In this case, the insurance company wants to encourage low-risk behavior at a cost of $100 for each individual. We need to find the deductible amount that would, on average, be equal to this cost.
Let's denote the deductible as D. The equation to calculate the deductible is:
D = (Probability of Getting HIV * Cost of HIV-related Illness) - (Probability of Not Getting HIV * Cost of Not Having HIV-related Illness)
In this case, the cost of HIV-related illness is $100,000, and the cost of not having HIV-related illness is $0. The probability of getting HIV is 2% (or 0.02) and the probability of not getting HIV is 98% (or 0.98).
Substituting these values into the equation, we get:
D = (0.02 * $100,000) - (0.98 * $0)
D = $2,000 - $0
D = $2,000
Therefore, the deductible must be $2,000 to encourage low-risk behavior.
d. To calculate the premium of the third policy, we need to consider that it does not cover HIV- or AIDS-related illnesses. Therefore, we only need to consider the expected cost of medical care without HIV.
The expected cost of medical care without HIV is $500.
The premium of the third policy would be the expected cost of medical care without HIV, which is $500.
Therefore, the premium of the third policy would be $500.