Quantitative Risk Management
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 ST 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 τ = 30%. The risk-free rate is RF = 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 ST . In particular, if we write it as a + b · max{ST − 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 units of 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 present value of the
hedged levered firm’s after-tax cash flows? How does it compare with your answer above, and
why?
