1.You are an analyst working for Goldman Sachs, and you are trying to value the growth potential of a large, established company, Big Industries. Big Industries has a thriving R&D division that has consistently turned out successful products. You estimate that, on average, the R&D division generates two new product proposals every three years, so that there is a 66% chance that a project will be proposed every year. Typically, the investment opportunities the R&D division produces require an initial investment of $10 million and yield profits of $1 million per year that grow at one of three possible growth rates in perpetuity: 3%, 0%, and −3%. All three growth rates are equally likely for any given project. These opportunities are always “take it or leave it” opportunities: If they are not undertaken immediately, they disappear forever. Assume that the cost of capital will always remain at 12% per year. What is the present value of all future growth opportunities Big Industries will produce?

2.Repeat Problem 11, but this time assume that all the probabilities are risk-neutral probabilities, which means the cost of capital is always the risk-free rate and risk-free rates are as follows: The current interest rate for a risk-free perpetuity is 8%; in one year, there is a 64.375% chance that all risk-free interest rates will be 10% and stay there forever, and a 35.625% chance that they will be 6% and stay there forever. The current one-year risk-free rate is 7%.

3.You own a small networking startup. You have just received an offer to buy your firm from a large, publicly traded firm, JCH Systems. Under the terms of the offer, you will receive 1 million shares of JCH. JCH stock currently trades for $25 per share. You can sell the shares of JCH that you will receive in the market at any time. But as part of the offer, JCH also agrees that at the end of the next year, it will buy the shares back from you for $25 per share if you desire. Suppose the current one-year risk-free rate is 6.18%, the volatility of JCH stock is 30%, and JCH does not pay dividends.

a. Is this offer worth more than $25 million? Explain.

b. What is the value of the offer?

4.You own a wholesale plumbing supply store. The store currently generates revenues of $1 million per year. Next year, revenues will either decrease by 10% or increase by 5%, with equal probability, and then stay at that level as long as you operate the store. You own the store outright. Other costs run $900,000 per year. There are no costs for shutting down; in that case, you can always sell the store for $500,000. What is the business worth today if the cost of capital is fixed at 10%?

5.You own a copper mine. The price of copper is currently $1.50 per pound. The mine produces 1 million pounds of copper per year and costs $2 million per year to operate. It has enough copper to operate for 100 years. Shutting the mine down would entail bringing the land up to EPA standards and is expected to cost $5 million. Reopening the mine once it is shut down would be an impossibility given current environmental standards. The price of copper has an equal (and independent) probability of going up or down by 25% each year for the next two years and then will stay at that level forever. Calculate the NPV of continuing to operate the mine if the cost of capital is fixed at 15%. Is it optimal to abandon the mine or keep it operating?

6.An original silver dollar from the late eighteenth century consists of approximately 24 grams of silver. At a price of $.019 per gram ($6 per troy ounce), the silver content of the coin is currently worth about $4.50. Assume that these coins are in plentiful supply and are not collector’s items, so they have no numismatic value. If the current price of silver is $0.19 per gram, will the price of the coin be greater than, less than, or equal to $4.50? Justify your answer.

7.What implicit assumption is made when managers use the equivalent annual benefit method to decide between two projects with different lives that use the same resource?

8.You own a cab company and are evaluating two options to replace your fleet. Either you can take out a five-year lease on the replacement cabs for $500 per month per cab, or you can purchase the cabs outright for $30,000, in which case the cabs will last eight years. You must return the cabs to the leasing company at the end of the lease. The leasing company is responsible for all maintenance costs, but if you purchase the cabs, you will buy a maintenance contract that will cost $100 per month for the life of each cab. Each cab will generate revenues of $1000 per month. Assume the cost of capital is fixed at 12%.

a. Calculate the NPV per cab of both possibilities: purchasing the cabs or leasing them.

b. Calculate the equivalent monthly annual benefit of both opportunities.

c. If you are leasing a cab, you have the opportunity to buy the used cab after five years. Assume that in five years a five-year-old cab will cost either $10,000 or $16,000 with equal likelihood, will have maintenance costs of $500 per month, and will last three more years. Which option should you take?