Lizhong is considering investing three quarters of his current income with Charles Schwab and the investment he is offered has an uncertain payoff. In particular, the investment manager tells Lizhong that the initial investment doubles in value 40% of the time, but one third of the initial investment is lost the other 60% of the time. Furthermore, Charles Schwab’s investment manager charges a fixed fee of F = 4 in order to process the investment transaction. Lizhong's current income is M = 40 and his utility function defined over income is U(M) = 1 - 1/M.
a. What is the expected value of the investment?
b. Demonstrate that under the current terms, Lizhong will choose not to make the investment. Carry your calculation out to three decimal places.
c. If the investment manager is risk neutral, can he offer Lizhong a fee schedule that will induce him to make the investment? If so, what is the fee schedule? If not, why isn’t this possible?
d. What is the certainty equivalent of the investment described in part c?