Suppose the preferences of some person can be represented by an EUF with NMUF µ(x)=√x suppose the person has initial wealth x0=1000 and he holds a lottery that gives him additional gain and losses =0.5(1000)+0.5(-1000) how much would that person be willing to pay for insurance against losses?
What is the risk premium for this lottery?
To determine how much the person would be willing to pay for insurance against losses, we need to find the amount at which the person is indifferent between taking the lottery and paying for insurance.
We can start by calculating the expected utility of the lottery without insurance:
EU(lottery) = 0.5 * µ(1000 + 1000) + 0.5 * µ(1000 - 1000)
= 0.5 * √(2000) + 0.5 * √(0)
= 0.5 * √(2000)
= 0.5 * 44.72
= 22.36
Now let's calculate the expected utility with insurance. Let's assume the person pays a premium (P) for the insurance. In case of a loss, the insurance fully covers the loss, so the person's wealth after the lottery would be x0 = 1000 - P. In case of a gain, the person receives the gain (1000) minus the premium paid. The expected utility with insurance can be represented as:
EU(insurance) = 0.5 * µ(1000 - P + 1000) + 0.5 * µ(1000 - P - 1000)
= 0.5 * √(2000 - 2P) + 0.5 * √(0 - 2P)
= 0.5 * √(2000 - 2P)
= 0.5 * √(2000 - 2P)
To find the amount at which the person is indifferent between taking the lottery and paying for insurance, we set EU(lottery) equal to EU(insurance) and solve for P:
22.36 = 0.5 * √(2000 - 2P)
Squaring both sides to eliminate the square root:
(22.36)^2 = (0.5)^2 * (2000 - 2P)
498.9696 = 500 - P
P = 500 - 498.9696
P ≈ 1.03
Therefore, the person would be willing to pay approximately $1.03 for insurance against losses.
The risk premium for this lottery is the difference between the expected value of the lottery without insurance and the expected value of the lottery with insurance:
Risk Premium = EU(lottery) - EU(insurance)
= 22.36 - (0.5 * √(2000 - 2P))
= 22.36 - (0.5 * √(2000 - 2 * 1.03))
= 22.36 - (0.5 * √(1997.94))
≈ 22.36 - (0.5 * 44.67)
≈ 22.36 - 22.34
≈ 0.02
The risk premium for this lottery is approximately $0.02.