The Ministry of Natural Resources of a foreign country has asked Mominco, a large international minerals company, to consider bidding for the exploration and mining rights for a large tract of land in that country. In conversations with the Ministry, Mominco has been told that a cash bid of $90 million would be "nearly certain" to win Mominco these rights. Based on this information and on experience in dealing with such situations before, Mominco’s management believes that a bid of $90 million would have a probability of 0.95 of winning the rights. However, based on the current financial needs of the country’s government and the likely levels of interest of competitors in this tract, management also believes that a much lower bid would not necessarily be turned down. Consequently, Mominco is also considering two other possible bid levels, $40 million and $65 million, as well as not bidding at all. Management believes these bids would have probabilities of 0.33 and 0.75, respectively, of winning the rights to this tract.
Mominco’s management already knows that a valuable mineral deposit lies
within the tract, but the size of the deposit remains highly uncertain.
Sensitivity analysis has shown that the size of the deposit is the crucial
uncertainty in evaluating the tract’s profitability — i.e., it is much
more important than the other uncertainties present. Based on available
data and on their experience and judgment, Mominco’s geologists have used
standard probability assessment methods to come up with the cumulative
probability distribution for the size of the deposit, which is shown in
the figure below.
In order to use this information in a decision tree, it is necessary to approximate the continuous distribution shown with a discrete distribution. The extended Pearson-Tukey approximation is suitable for that purpose. It utilizes the 0.05, 0.50, and 0.95 fractiles (5th, 50th, and 95th percentiles) from the underlying continuous distribution as the three possible values in the approximating discrete distribution and assigns them probabilities of 0.185, 0.630, and 0.185, respectively. Note that these three fractiles correspond to 20, 40, and 80 million tons in this case.
Mominco’s planning staff has determined that V, the net present value (NPV) to Mominco from developing the tract after taxes, royalties, etc., but excluding the bid, can be represented by
A. Draw an influence diagram for Mominco’s decision problem that shows the variables (decisions and uncertainties), the evaluation measure, and the relationships among them. You need not include numerical data at this stage.
B. Draw a neatly labeled decision tree representing Mominco’s decision problem. Include probabilities and branch values throughout the tree and a value for the evaluation measure (NPV) at each endpoint to represent the overall consequences of following that path.
C. Assuming management wishes to use expected NPV as the criterion for decision making, solve the decision tree from part B and show the optimal strategy on it by indicating the optimal alternative (or crossing out all non-optimal alternatives) at each decision node. Please show the expected value at each decision and probability node so that the steps in your solution can be followed (you need not show all of the arithmetic.)
D. Express the optimal strategy you found in part C verbally: i.e., describe the best initial choice and the best choices for subsequent decisions that may arise (if any). Also, specify the corresponding (optimal) expected NPV verbally.
E. Find the probability distribution (PDF, or PMF), or "risk profile," corresponding to the optimal strategy and present it in tabular form.
F. Management is concerned about the sensitivity of the decision in part C to the probability, hereafter denoted by p, of winning the rights with a $40 million bid. Find the "critical value" of p (to two decimal places) at which the optimal strategy in part C changes and describe the new optimal strategy. Does this change in strategy make sense? Explain.
G. Management is also concerned about the effects of risk aversion on the optimal strategy from part C. Assume Mominco’s risk preferences for NPV can be represented by an exponential utility function of the form -exp(-x/r ), where x represents NPV and r is Mominco’s risk tolerance. What is Mominco’s optimal strategy if its risk tolerance is $25 million and what is the corresponding certainty equivalent? At what level of risk tolerance (to the nearest $1 million) does the optimal strategy change from that in part C to the strategy you just found? Explain intuitively why this change occurs. Are there additional changes in strategy as risk tolerance decreases below $25 million? Explain.