Fred has $W to invest. He is considering investing in a defense company, Zee Co. If Zee Co is awarded a new defense contract, then the value of any...

3. Fred has $W to invest. He is considering investing in a defense company, Zee Co. If Zee Co is awarded a new defense contract, then the value of any investment in Zee Co will double. However, if it does not receive a new contract then the value of the investment will be cut in half. Fred believes that it is equally likely that Zee Co will receive the new contract. Finally assume that his preferences over his investment (x) can be represented by the utility function u(x) = ln x.

For parts (a) - (c) Assume that Fred has put all of his money in Zee Co (so he has invested $W in Zee Co).

a) What is the expected value of his investment (as a function of W)? What is his expected utility of his investment?

b) What is the certainty equivalent to his investment (as a function of W)? What is the risk premium?

c) Illustrate his utility function and the expected utility of his investment in a diagram. Label the expected value, the certainty equivalent and the risk premium.

d) If his only options are investing all of his money in Zee Co or keeping his money in cash (where it will have a certain value of $W) which (if any) would he prefer? 

For parts (e) - (g) below assume that Fred may put any fraction, α, of his $W in Zee Co. The remaining fraction, (1-α)W, will be kept in cash. Assume that the returns on money put in Zee Co and the associated probabilities are as above.

e) Write his expected utility function as a function of α. Write the FOC associated with Fred's expected utility maximization problem.

f) Show that at α=0 the first derivative of expected utility is positive (i.e. 0 does not solve the FOC). Briefly explain why this is not a surprising result.

g) What is the value of α that maximizes his expected utility?