Chapter22: Real Options
22-3 The Abandonment Option
Expansion value is important. When investments turn out well, the quicker and easier the business can be expanded, the better. But suppose bad news arrives, and cash flows are far below expectations. In that case it is useful to have the option to bail out and recover the value of the project's plant, equipment, or other assets. The option to abandon is equivalent to a put option. You exercise that abandonment option if the value recovered from the project's assets is greater than the present value of continuing the project for at least one more period.
The binomial method is tailor-made for most abandonment options. Here is an example.
The Zircon Subductor Project
Dawn East, the chief financial officer of Maine Subductor Corp., has to decide whether to start production of zircon subductors. The investment required is $12 million—$2 million for roads and site preparation and $10 million for equipment. The equipment costs $700,000 per year ($.7 million) to operate (a fixed cost). For simplicity, we ignore other costs and taxes.p. 562
At today's prices, the project would generate revenues of $2.5 million per year. Annual output will be constant, so revenue is proportional to price. If the mine were operating today, cash flow would be $2.5 − .7 = $1.8 million.
Calculate the Present Value of the Project The first step in a real options analysis is to value the underlying asset, that is, the project if it had no options attached. Usually this is done by discounted cash flow (DCF). In this case the chief source of uncertainty is the future selling price of zircon subductors. Therefore Ms. East starts by calculating the present value of future revenues. She perceives no upward trend in subductor prices, and ends up forecasting stable prices for the next 8 years. Fixed costs are constant at $.7 million. The top panel of Figure 22.4 shows these cash-flow forecasts and calculates present values: about $13.8 million for revenues, after discounting at a risk-adjusted rate of 9%, and $4.3 million for fixed costs, after discounting at a risk-free rate of 6%.7 The NPV of the project, assuming no salvage value or abandonment over its 10-year life, is:
This NPV is slightly negative, but Ms. East has so far made no provision for abandonment.
Build a Binomial Tree Now Ms. East constructs a binomial tree for revenues and PV(revenues). She notes that subductor prices have followed a random walk with an annual standard deviation of about 20%. She constructs a binomial tree with one step per year. The “up” values for revenues are 122% of the prior year's revenues. The “down” values are 82% of prior revenues.8 Thus, the up and down revenues for year 1 are $2.5 × 1.22 = $3.05 and $2.5 × .82 = $2.05 million, respectively. After deduction of fixed costs, the up and down cash flows are $2.35 and $1.35 million, respectively. The first two years of the resulting tree are shown below (figures in millions of dollars).p. 563
Figure 22.4 shows the whole tree, starting with cash flows in year 1. (Maine Subductor can't generate any revenues in year 0 because it hasn't started production yet.) The top number at each node is cash flow. The bottom number is the end-of-year present value of all subsequent cash flows, including the value of the production equipment when the project ends or is abandoned. We will see in a moment how these present values are calculated.
Finally, Ms. East calculates the risk-neutral probabilities of up and down changes in revenues, p and 1 − p, respectively. Here she must pause to make sure that each year's revenue is valued properly. Remember that we have discounted revenues at a risk-adjusted rate of 9%. Thus the present value of year 1 revenues is not $2.5 million, but only
Therefore, Ms. East needs to calculate the risk-neutral probabilities that generate an expected return equal to the 6% risk-free rate.9
Ms. East can use these probabilities at every node of the binomial tree, because the proportional up and down moves are the same at each node.
Solve for Optimal Abandonment and Project Value Ms. East has assumed a project life of 8 years. At that time the production equipment, which normally depreciates by about 5% per year, should be worth $6.63 million. This salvage value represents what the equipment could be sold for, or its value to Maine Subductor if shifted to another use.
Now let's calculate this project's value in the binomial tree. We start at the far right of Figure 22.4 (year 8) and work back to the present. The company will abandon for sure in year 8, when the ore body is exhausted. Thus we enter the ending salvage value ($6.63 million) as the end-of-year value in year 8. Then we step back to year 7.
Suppose that Maine Subductor ends up in the best possible place in that year, where cash flow is $9.44 million. The upside payoff if the company does not abandon is the “up” node in year 8: 11.68 + 6.63 = $18.31 million. The downside payoff is 7.60 + 6.63 = $14.23 million. The present value, using the risk-neutral probabilities, is:p. 564
The company could abandon at the end of year 7, realizing salvage value of $6.98 million, but continuing is better. We therefore enter $14.90 million as the end-of-year value at the top node for year 7 in Figure 22.4.
We can fill in the end-of-period values for the other nodes in year 7 by the same procedure. But at some point, as we step down to lower and lower cash flows, there will come a node where it's better to bail out than continue. This occurs when cash flow is $.67 million. The present value of continuing is only:
The cash flows and end-of-year values for year 7 are the payoffs to continuing from year 6. We then calculate values in year 6, checking at each node whether to abandon. Repeat this drill for year 5, then year 4, and so on back to year 0. In this example, Maine Subductor should abandon the project if cash flows drop to $.67 million or less in each year. We have colored the nodes in Figure 22.4 where abandonment occurs.
Solving back through the binomial tree, we get a present value of $13.977 million at year 0, and net present value of $13.977 − 12.0 = $1.977 million.10 If there were no option to abandon, the DCF valuation would be − $2.51 million. Therefore the option to abandon is worth $1.977 + 2.510 = $4.487 million.11 In an APV format,
Abandonment Value and Project Life
Ms. East assumed that the zircon subductor project had a definite 8-year life. But most projects' economic lives are not known at the start. Cash flows from a new product may last only a year or so if the product fails in the marketplace. But if it succeeds, that product, or variations or improvements of it, could be produced for decades.
A project's economic life can be just as hard to predict as the project's cash flows. Yet in standard DCF capital-budgeting analysis, that life is assumed to end at a fixed future date. Real options analysis allows us to relax that assumption. Here is the procedure.12
This procedure links project life to the performance of the project. It does not impose an arbitrary ending date, except in the far distant future.
Companies are often faced with complex options that allow them to abandon a project temporarily, that is, to mothball it until conditions improve. Suppose you own an oil tanker operating in the short-term spot market. (In other words, you charter the tanker voyage by voyage, at whatever short-term charter rates prevail at the start of the voyage.) The tanker costs $50 million a year to operate and at current tanker rates it produces charter revenues of $52.5 million per year. The tanker is therefore profitable but scarcely cause for celebration. Now tanker rates dip by about 10%, forcing revenues down to $47 million. Do you immediately lay off the crew and mothball the tanker until prices recover? The answer is clearly yes if the tanker business can be turned on and off like a faucet. But that is unrealistic. There is a fixed cost to mothballing the tanker. You don't want to incur this cost only to regret your decision next month if rates rebound to their earlier level. The higher the costs of mothballing and the more variable the level of charter rates, the greater the loss that you will be prepared to bear before you call it quits and lay up the boat.
Suppose that eventually you do decide to take the boat off the market. You lay up the tanker temporarily.13 Two years later your faith is rewarded; charter rates rise, and the revenues from operating the tanker creep above the operating cost of $50 million. Do you reactivate immediately? Not if there are costs to doing so. It makes more sense to wait until the project is well in the black and you can be fairly confident that you will not regret the cost of bringing the tanker back into operation.
These choices are illustrated in Figure 22.5. The blue line shows how the value of an operating tanker varies with the level of charter rates. The black line shows the value of the tanker when mothballed.14 The level of rates at which it pays to mothball is given by M and the level at which it pays to reactivate is given by R. The higher the costs of mothballing and reactivating and the greater the variability in tanker rates, the further apart these points will be. You can see that it will pay for you to mothball as soon as the value of a mothballed tanker reaches the value of an operating tanker plus the costs of mothballing. It will pay to reactivate as soon as the value of a tanker that is operating in the spot market reaches the value of a mothballed tanker plus the costs of reactivating. If the level of rates falls below M, the value of the tanker is given by the black line; if the level is greater than R, value is given by the blue line. If rates lie between M and R, the tanker's value depends on whether it happens to be mothballed or operating.
7Why calculate present values for revenues and fixed costs separately? Because it's easier to construct a binomial tree for revenues, which can be assumed to follow a random walk with constant standard deviation. We will subtract fixed costs after the binomial tree is constructed.
8The formula (given in Section 22.2) for the up percentage is
9Notice that the “up” revenues are 122% of today's revenue level, but 133% of the present value of next year's forecasted revenues. Thus the “up” probability required to generate a 6% average return is relatively small.
11It turns out, however, that the value of early abandonment in this example is relatively small. Suppose that Maine Subductor could recover salvage value of $6.63 million in year 8, but not before. The present value of this recovery in year 0, using a 6% discount rate, is $4.16 million. APV in this case is − 2.51 + 4.16 = $1.65 million, a little less than the APV of $1.977 million if early abandonment is allowed.
12See S. C. Myers and S. Majd, “Abandonment Value and Project Life,” in Advances in Futures and Options Research, ed. F. J. Fabozzi (Greenwich, CT: JAI Press, 1990).
13We assume it makes sense to keep the tanker in mothballs. If rates fall sufficiently, it will pay to scrap the tanker.
14Dixit and Pindyck estimate these thresholds for a medium-sized tanker and show how they depend on costs and the volatility of freight rates. See A. K. Dixit and R. S. Pindyck, Investment under Uncertainty (Princeton, NJ: Princeton University Press, 1994), Chapter 7. Brennan and Schwartz provide an analysis of a mining investment that also includes an option to shut down temporarily. See M. Brennan and E. Schwartz, “Evaluating Natural Resource Investments,” Journal of Business 58 (April 1985), pp. 135–157.