Consumer A has a utility function of U (w) in dollar. And she has a current wealth of $80,000. Consumer B has a utility function of U(w) = ,1-0.6 (106) w where w is her wealth = (1-10.4) w1-0.4 with a current wealth of 20,000. An investment opportunity offers a 60% chance of success with a payoff of $35,000 and pays nothing if the investment fails. The required investment outlay is $20,000.
c. An insurance agent offers an insurance plan to B. The plan will pay 3,000 if the investment succeeds, and 38,000 if it fails. Show that without considering the investment opportunity, the maximum price that the insurance agent can charge consumer B (ignore the time value of money) is $14,401.87. Will the insurance company be willing to offer this insurance plan to B at the maximum price (Hint: think about the expected profit of the company)?
d. Consider the proposed insurance in the previous part. The premium of the insurance is $17,000. Suppose B can borrow and lend at a risk-free interest rate of 2%, compounded continuously. The investment outlay and the insurance premium have to be paid today, if any. The investment takes one year to yield any results. How should B choose if she cares only about her welfare 1 year from today? (i.e. should she make the investment only, or make the investment and purchase insurance, or purchase the insurance only, or do nothing?) (10 points) Explain with supporting calculations. Note: if B chooses to do both, she has to borrow 17000, and 1 year later, she needs to repay 17000exp(2%); if she chooses to buy the insurance only, or do nothing, she can save(lend) her wealth at a rate of 2% for one year and collect interest.
(34 points) Consumer A has a utility function of U(w)=(10.61)w10.6 where w is her wealth in dollar. And she has a current wealth of $80,000. Consumer B has a utility function of U(w)= (10.41)w10.4 with a current wealth of 20,000 . An investment opportunity offers a 60% chance of success with a payoff of $35,000 and pays nothing if the investment fails. The required investment outlay is $20,000. a. Which consumer is more risk averse? Support your conclusion with calculations. Does your answer depend on different measures of risk aversion? If so, which measure should you use? Explain carefully with the corresponding derivation of your measures. (8 points) b. Based on B's preference, should she make the investment? Suppose B does make the investment. Find the B's certainty equivalent and risk premium in this scenario. (8 points) c. An insurance agent offers an insurance plan to B. The plan will pay 3,000 if the investment succeeds, and 38,000 if it fails. Show that without considering the investment opportunity, the maximum price that the insurance agent can charge consumer B (ignore the time value of money) is $14,401.87. Will the insurance company be willing to offer this insurance plan to B at the maximum price (Hint: think about the expected profit of the company)? (8 points) d. Consider the proposed insurance in the previous part. The premium of the insurance is $17,000. Suppose B can borrow and lend at a risk-free interest rate of 2%, compounded continuously. The investment outlay and the insurance premium have to be paid today, if any. The investment takes one year to yield any results. How should B choose if she cares only about her welfare 1 year from today? (i.e. should she make the investment only, or make the investment and purchase insurance, or purchase the insurance only, or do nothing?) (10 points) Explain with supporting calculations. Note: if B chooses to do both, she has to borrow 17000 , and 1 year later, she needs to repay 17000exp(2%); if she chooses to buy the insurance only, or do nothing, she can save(lend) her wealth at a rate of 2% for one year and collect interest. ( 34 points) Consumer A has a utility function of U(w)=(10.61)w10.6 where w is her wealth in dollar. And she has a current wealth of $80,000. Consumer B has a utility function of U(w)= (10.41)w10.4 with a current wealth of 20,000 . An investment opportunity offers a 60% chance of success with a payoff of $35,000 and pays nothing if the investment fails. The required investment outlay is $20,000. a. Which consumer is more risk averse? Support your conclusion with calculations. Does your answer depend on different measures of risk aversion? If so, which measure should you use? Explain carefully with the corresponding derivation of your measures. (8 points) b. Based on B's preference, should she make the investment? Suppose B does make the investment. Find the B's certainty equivalent and risk premium in this scenario. (8 points) c. An insurance agent offers an insurance plan to B. The plan will pay 3,000 if the investment succeeds, and 38,000 if it fails. Show that without considering the investment opportunity, the maximum price that the insurance agent can charge consumer B (ignore the time value of money) is $14,401.87. Will the insurance company be willing to offer this insurance plan to B at the maximum price (Hint: think about the expected profit of the company)? (8 points) d. Consider the proposed insurance in the previous part. The premium of the insurance is $17,000. Suppose B can borrow and lend at a risk-free interest rate of 2%, compounded continuously. The investment outlay and the insurance premium have to be paid today, if any. The investment takes one year to yield any results. How should B choose if she cares only about her welfare 1 year from today? (i.e. should she make the investment only, or make the investment and purchase insurance, or purchase the insurance only, or do nothing?) (10 points) Explain with supporting calculations. Note: if B chooses to do both, she has to borrow 17000 , and 1 year later, she needs to repay 17000exp(2%); if she chooses to buy the insurance only, or do nothing, she can save(lend) her wealth at a rate of 2% for one year and collect interest
