Q1. (Sensitivity Analysis) The town of Kingston, NY is considering building a health & wellness center. The estimated construction cost is $9 million with annual staffing and maintenance costs of $750,000 over the twenty year life of the project. At the end of the life of the project, Kingston expects to be able to sell the land for $3 million, though the amount could be as low as $1 million and as high as $5 million.
Analysts estimate the first year benefits (accruing at the end of the year of the first year) to be $1.2 million. They expect the annual benefit to grow in real terms due to increases in population and income.
Their prediction is a growth rate of 2 percent, but it could be as low as 0 percent and as high as 4 percent. Analysts estimate the real discount rate for Kingston to be 5 percent, though they acknowledge that it could be a percentage point higher or lower.
(a) Calculate the present value of net benefits for the project using the analysts' predictions.
(b) Investigate the sensitivity of the present value of net benefits to alternative predictions within the ranges given by the analysts.
Q2. (Expected Value Analysis) The initial cost of constructing a permanent dam (i.e., a dam that is expected to last forever) is $513 million. The annual net benefits will depend on the amount of rainfall: $15 million in a "dry" year, $36 million in a "wet" year, and $45 million in a "flood" year. Meteorological records indicate that over the last 100 years there have been 75 "dry" years, 22 "wet" years, and 3 "flood" years. Assume the annual benefits, measured in real dollars, begin to accrue at the end of the first year. Use the meteorological record as a basis for assigning probabilities to "dry", "wet" and "flood" years. Assume a real discount rate of 5 percent.
(a) Calculate the expected value of the annual net benefits.
(b) Calculate the present value of the stream of (expected) annual net benefits. (HINT: The formula for the present value of perpetuity is
PV of perpetuity = Annual Payment/Discount Rate
(c) What are the net benefits of the dam?
(d) What is the internal rate of return of the dam? (HINT: Try nearby whole number discount rates).
Q3. (Monte Carlo analysis) An online apparel retailer will sell winter coats this season. The retailer purchases the coats from a supplier at a cost of $175 and will sell them for $250. Demand has been forecasted to be 2,000 coats this season, but is not known for certain and could range anywhere from 1,000 to 3,000 coats. At the end of the season, the retailer will have a 1/2 off sale in order to clear out the inventory.
(a) Assume a uniform distribution for demand. Assume the retailer orders 2,000 coats. Construct a simulation of 1,000 trials to evaluate the distribution of profit. What is average profit over your trials? What is the confidence interval within which profit is likely to fall 95% of the time?
(b) Graph the distribution of profit in your simulation. How does it differ from the uniform distribution?
(c) Simulate order sizes in 50-unit increments from 1,500 coats to 2,500 coats. Determine the order size that appears to maximize expected profit. Is this order size equal to average demand? Why or why not?
Q4. (Quasi-option Value) Imagine that the net present value of a hydroelectric plant with a life of 70 years is $24 million and that the net present value of a thermal electric plant with a life of 35 years is $18.77 million. Assume a discount rate of 5%.
(a) Compute the net present value of rolling the thermal plant over twice to match the life of the hydroelectric plant.
(b) Now assume that at the end of the first 35 years, there will be an improved second 35-year plant. Specifically, there is a 25 percent chance that an advanced solar or nuclear alternative will be available that will increase the net benefits by a factor of three; a 70 percent chance that a major improvement in thermal technology will increase net benefits by 50 percent; and a 5 percent chance that more modest improvements in thermal technology will increase net benefits by 10 percent.
(a) Should the hydroelectric or thermal plant be built today?
(b) What is the quasi-option value of the thermal plant?
Q5. (Expected Utility) A worker currently makes $1,200 per month waiting tables in Alabama. His friend in New York tells him that there might be a temporary job for him in New York that pays more. But, he would have to come to New York now. If he comes, there is a 90% chance there will be a job paying x, and a 10% chance that he'll earn nothing for that month. (The next month he'll be back in Alabama either way.) Suppose his utility over income is given by the function u(x) = √x. What is the minimum this job would need to pay to convince him to take the chance coming to New York? Assume his transport costs are $150 for the return trip to New York. Ignore differences in cost of living between Alabama and New York. Assume that the worker has the $150 available to pay the transport cost.
Q6. (Option Price) Imagine that a rancher would have an income of $100,000 if his county remains free from a cattle parasite but only $50,000 if the county is exposed to the parasite. Imagine that a county program to limit the impact of exposure to the parasite would reduce his income to $93,000 if the county remains free of the parasite but increase it to $75,000 if the county is exposed to the parasite. Assume that there is a 25 percent chance of exposure to the parasite and that the rancher's utility is the natural logarithm of his income. What is the rancher's option price for the county program? You will find it useful to construct a spreadsheet to compute your solutions.