FACULTY OF BUSINESS AND ECONOMICS
MFIN7016
Real Options and Dynamic Corporate Finance
Module 5
Final Examination
Submit your answers to all three questions before 10:00 pm on April 23, 2025.
1. In December 2024, your company faces a decision to bid for a eight-year lease
on a coal mine offered by the government. The lease gives the winner of the auction the right to actually develop the mine for four years. Once development takes place (if ever at all), the leaseholder can extract 25 million tons of coal per year for exactly four years with the fixed extraction cost of $4 per ton. Development of the mine costs $50 million. There are no reclamation costs at the end of the life of the mine. If development did not take place by December 2028 (four years down the road), then the lease would end at that time.
At the time of the auction, the spot price of coal was $4.58. The risk-free rate of interest is 6% per year and the net convenience yield on coal is 3.5% per year, both of which are constant and compounded continuously. The standard deviation of log changes in coal prices is 15% per year. Development costs are depreciated on a straight-line basis over four years. There are no net working capital requirements. The marginal tax rate of your company is 40%. There is no inflation.
(a) Suppose that your company commits to an immediate development of the mine. What is the maximum amount that your company should bid in the auction for the lease?
(b) Build a four-step binomial tree with a time step set equal to one year. Calculate the value of the lease by the real options approach.
(c) Now, your company knows that the government might terminate the lease (if development has not occurred) at the beginning of the fourth year with a 10% chance. However, if development has started at or before the end of the third year, there is no risk of a government termination of the lease. Redo part (b). Why is the value of the lease different from that in part (b)?
2. Consider a risk-neutral firm that has a project whose value evolves over time according to the following geometric Brownian motion:
dV (t) = 0.02V (t)dt + 0.3V (t)dZ(t),
where the growth rate is 2% per year, the volatility is 30% per year, and dZ(t) is the increment of a standard Wiener process. The value of the project at t = 0 is $10 million. The instantaneous risk-free rate of interest is 5% per year. As such, the convenience yield is 5% - 2% = 3% per year. The firm can undertake the project at any time t by paying $10 million at that time to receive V (t).
(a) As an approximation, the firm employs a discrete-time analysis to deter- mine its optimal investment decision as well as the value of the investment option at t = 0. Specifically, the firm builds a four-step binomial tree to approximate the geometric Brownian motion of V (t), where each time step is of one year. In this framework, the firm assumes that the project only lasts for four years. What is the threshold value of the project (i.e., investment trigger) at which the firm undertakes the project? What is the value of the investment option at t = 0?
(b) Now the firm uses a continuous-time analysis, taking into account that the investment option is perpetual. Let V* be the threshold value of the project (i.e., the investment trigger) and F(V (t)) be the value of the investment option at time t. If V (t) ≥ V* (i.e., the stopping region), the project is immediately undertaken. If V (t) < V* (i.e., the continuation region), construct a portfolio that consists of the investment option and the project so as to establish a no-arbitrage relationship that governs F(V (t)). What is the resulting ordinary differential equation? Apply conditions to the boundaries of the continuation region, (0, V*), and solve the differential equation for the numerical values of F(V (0)) and V* , where V (0) = $10 million.
(c) Contrast the results in parts (a) and (b). How good is the approximation?
3. Why is it that the true NPV of real asset investments can be positive or negative,
while the true NPV of financial market investments is equal to zero?