Include workings.
2. Consider a firm that produces 10 units of gold a year from today. The price of gold next year, ST, is normally distributed with mean 100 and volatility of 20%. The firm knows that there will be a buyer who is willing to pay the price of 90 per unit, no matter what the value of S is. At T = 1, the firm can choose whether to sell to this buyer or at the market price of ST. Firm cash flows are taxed at a flat rate of 7 = 30%. The risk-free rate is Ap = 5% and is compounded annually. Use two decimal places for your answers. (a) [2 points] Express the before-tax cash flows of the (unhedged) firm as a function of the gold spot price Sp. In particular, if we write it as a + b . max{Sy -90,0}, what are the values of a and b? (b) [2 points] Now suppose that there is a call option on gold whose strike price is 90. The premium of this option is 17. The firm decides to sell this call option to perfectly hedge its cash flow risk. If the proceeds from selling these options are invested in the risk-free asset and added to the firm profit in one year, what is the before-tax payoff of the hedged firm in one year? (c) [1 point] What is the present value of the cash flows of the hedged unlevered firm after taxes? (d) [3 points] If the hedged firm above issues the maximum amount of safe debt D to take advantage of the fact that the interest payments are tax deductible, what is the total value of the hedged levered firm after tax? How does it compare with your answer above, and why?1. (35) Consider a variation of the Glosten-Milgrom sequential trade model where the asset's value V can take three values. Suppose that the true value of stock in Trident Corporation can be, with equal probability, either VH : %, VL : i, or some middle value VM. Let a: : % of the traders be informed insiders, while the remaining 1 o: : g are uninformed noise traders. Assume as always that informed traders always buy when V = VH and sell when V = VL, while uninformed traders buy or sell with equal probability. The focus of this problem is the traders' behavior when V 2 VM . (a) (5) Draw the tree diagram, leaving uncertain the action of informed traders when V 2 VM . (b) (5) Show that there is no value of VM for which informed traders randomize between buying and selling. (c) (10) Suppose that informed traders always buy when V 2 VM . i. (3) Calculate the conditional probabilities of a buy order at each value V can take and the uncondi- tional probability of a buy. ii. (3) Using Bayes' rule, calculate the posterior probabilities of V taking on each value conditional on a buy, and compute the ask price as a function of VM . iii. (4) Find the informed trader's payoff when V 2 VM and use this to nd the lowest value of VM at which the trader is willing to buy. (d) (10) Now suppose the informed traders always sell when V 2 VM . i. (3) Calculate the conditional probabilities of a sell order at each value V can take and the uncondi- tional probability of a sell. ii. (3) Using Bayes' rule, calculate the posterior probabilities of V taking on each value conditional on a sell, and compute the bid price as a function of VM . iii. (4) Find the informed trader's payoff when V 2 VM and use this to nd the highest value of VM at which the trader is willing to sell. (e) (5) What happens if VM satises neither of the bounds you found above? (2) Extemalities (10 pts) Like under Question (I), assume the private demand curve (private marginal utility) for automobile trips is given by P=ZUfU-3Q. The private supply curve (private marginal cost curve) W P stands for the price of gasoline and Q for vehicles miles driven. a) Calculate the private equilibrium b) Now assume the external cost instead of being constantly 50 per unit, it always equals 20% of the private MC. What is the socially optimal quantity? c) What is the DWL of the private solution? d) Assume the city were to impose a Pigou Tax on the supply side. What is the tax ($lunit) at the privately optimal quantity? What is the tax ($lunit) at the socially optimal quantity
