CHAPTER 12 APPENDIX Valuing Some More Real Options

CHAPTER 12 APPENDIX Valuing Some More Real Options This appendix demonstrates how to work out the value of different types of real options. By assuming the world is risk neutral, it is ignoring the fact that discount rates can be higher when there are more real options. (Depending on the context, not fretting too much about the correct discount rate can be forgivable or deadly.) Real options are tough enough to value even without this added complication. This is not an easy appendix! We ignore the problem of assigning different costs of capital to real options. 12.10 DECISION TREES: ONE SET OF PARAMETERS Assume that you own a firm that can produce 150,000 units of a good at a cost of $100/unit. The retail price of your good was $500/unit recently, but you now expect it to go up or down by $100/unit this year, that is, either to $400/unit or $600/unit. The year thereafter, you expect it to go up or down by $200/unit. These price scenarios can be shown in a simple tree: Time: Recent 1 2 $800 $600 $400 $500 $600 $400 $200 All price changes are equally likely. The fixed costs of running the plant are $50 million, and rent (regardless of whether you run the plant or not) is $10 million. The world is risk neutral and the prevailing interest rate is 10% per year, which applies to this coming year s cash flows and which will be twice compounded when applied to the following year s cash flows. Moreover, assume that you know at the beginning of each year what the price over the whole year will be, because you receive customer orders at this point. (To model intrayear uncertainty more realistically, you would have to deal with more periods not any more difficult in concept but much more tedious.) As an example, compute the firm value if you know that the price will go to $600/unit and then to $400/unit, and if you know that you will operate the plant this year but not the following year. The first year, you would earn revenues of 150,000 units. ($600/unit $100/unit) = $75 million, pay fixed costs of $50 million, and rent of $10 million. Your net profits would be $15 million, which discounts to $13.64 million at 10% if you use the present value formula. The second year, you would earn no revenues and pay no fixed costs, but you would still pay rent of $10 million. This discounts to $10/1.1 2 $8.26 million. In sum, under this price path and with this operating policy, your firm would have an NPV of $13.64 $8.26 = $5.38 million. Let s take the same project and consider its value in a number of scenarios, which differ in the assumption of what you know and how you can respond to the prevailing environment. A more involved two-level tree example. The cost of capital and timing assumptions. Here is an illustration of how the model works. Your task is to work out value based on your ability to respond to the environment. 433

434 CHAPTER 12 APPENDIX VALUING SOME MORE REAL OPTIONS I rigged the example the firm value is $0 if you have no flexibility. If you have perfect flexibility, you get the max. No flexibility all choices made up front: First, let s compute the value under inflexible behavior. This is one extreme benchmark. What is the value if you have to make your decision today of whether to operate or not in all future scenarios? That is, the firm would either have to operate or not operate in both future periods with the $600/unit and $400/unit scenarios. If you do not start the plant, you would simply value the firm at $0. If you do start the plant, then you must make the calculations that the tree in Figure 12.2 shows. If the price increases to $600, you earn $75 $50 $10 = $15 million. If it decreases to $400, you earn $45 $10 $50 = $15 million. Therefore, your expected revenues are $0. The following year, you earn +$45 million, $15 million, +$15 million, or $45 million. This again comes to an expected $0. In this example, it really does not matter whether you start the plant or not your firm value is always $0. Importantly, this $0 is also the value if you work with expected outcomes instead of the tree. The expected price in both future years is $500/unit. At the expected price, your $100/unit production cost translates into expected revenues of $60 million. You would still have to pay for rent and fixed costs, at $60 million per year. Indeed, working with expected values is the same as assuming that you do not have the ability to make strategic choices in the future (discussed next) a common source of underestimated project values in practice. All real options the fully flexible choice: Now assume the opposite extreme benchmark: You know each year what the price is and you have perfect flexibility to shut down and reopen the plant in response to market conditions. This option is called the timing option. Here, if the retail price is above $500/unit, you would operate. For example, if the retail price is $600/unit, your marginal revenues are $150,000. ($600/unit $100/unit) $50,000,000 = $25,000,000. Subtract $10 million in sunk rent cost, and you end up with revenues of $15 million. If the retail price is $400/unit, you earn $45 million, which is not enough to cover the $50 million fixed operating costs, so you are better off not operating and just paying the rent of $10 million. Figure 12.3 shows your valuation and optimal decision tree now. Again, the figure highlights important flexibility-related choices in blue. The heavy boxes indicate that you operate the plant; the other boxes indicate that you do not. You earn +$15million or $10million in the first year. The expected value is $2.5million, which discounts to $2.3 million (indicated at the bottom of the figure). The final year, you earn +$45 million, $10 million, +$15 million, or $10 million, which is an expected value of $10 million and a discounted value of $8.3 million. Therefore, this firm is worth about +$10.5 million. The value to having knowledge and the flexibility to act on it (knowledge without flexibility is useless!) has transformed this firm from a nothing into a gem. It is this value-through-flexibility that your strategic option to respond has created. Put differently, the value of your real option is +$10.5 million. The option to delay choice: Often, you do not have full flexibility. Instead, you have some real options, but not perfect flexibility. For example, what would happen

12.10 DECISION TREES: ONE SET OF PARAMETERS 435 Flexibility: Plant (or not) Net: $15,000,000 Net: $15,000,000 Revenues: $15,000,000 Net: $45,000,000 NPV = $0M PV(E(C 1 )) = $0M PV(E(C 2 )) = $0M FIGURE 12.2 Value Under No Flexibility Always Operate the Plant if you had the option to delay your decision by 1 year, more specifically, to run the plant only if the price appreciates to $600/unit, but not if it depreciates to $400/unit? If you run the plant next year, you have to run it the following year. If you do not run the plant next year, you cannot run it the following year, either. Figure 12.4 shows your revised decision tree. The average outcome is $5 million divided by 2 in the first year, and $10 million divided by 4 in the second year. Discount the first by 10% and the second by 21%, and you find the net of $2.5/1.1 + $2.5/1.1 2 $4.3 million. You can come to the same $4.3 million solution by following your decisions in time:

436 CHAPTER 12 APPENDIX VALUING SOME MORE REAL OPTIONS Flexibility: Plant (or not) Flexibility: Plant (or not) Flexibility: (Plant or) Not Flexibility: (Plant or) Not Flexibility: Plant (or not) Flexibility: (Plant or) Not NPV $10.537M PV(E(C 1 )) = $2.5/1.1 $2.273M PV(E(C 2 )) = $10/1.1 2 $8.264M FIGURE 12.3 Value Under Perfect Flexibility Full Knowledge and Choice If the retail price increases to $600/unit, your best decision is to operate the plant. You will earn $15 million in the first year, and either gain $45 million or lose $15 million the second year. Your net is $15/1.10 $13.6 million plus (0.5. $45 + 0.5. [ $15])/1.10 2 $12.4 million. The total is $26 million in expected present value. If the retail price falls to $400, you commit to shuttering the plant. Your net is a sure loss of $10 million in each of the 2 years. In present value, this is $9.1

12.10 DECISION TREES: ONE SET OF PARAMETERS 437 Flexibility: Commit fully Decision: None (plant runs) Decision: None (plant runs) Net: $15,000,000 Flexibility: Abandon fully Decision: None (plant closed) Decision: None (plant closed) NPV $4.339M PV(E(C 1 )) = $2.5/1.1 $2.273M PV(E(C 2 )) = $2.5/1.1 2 $2.066M FIGURE 12.4 Value to 1-Year-Ahead Information (or Ability to Delay Choice until Year 1) million followed by $8.3 million. Your total is a loss of $17.4 million in expected present value. Both price paths are equally likely, so the plant is worth about 0.5. ( $17.4) + 0.5. $26 $4.3 million. Intuitively, the reason why a plant with this more limited real option does not reach +$10.5 million under the full flexibility real option is that you would still have to operate the plant in the final period if the price is $400/unit (which you would rather not do), and you would fail to run the plant in the final period if the price is $600/unit (which you would rather do). The option to start later: An alternative scenario would allow you to start the plant anytime you wish, but once you start the plant, you cannot stop it. Figure 12.5

438 CHAPTER 12 APPENDIX VALUING SOME MORE REAL OPTIONS Flexibility: Commit fully Decision: None (plant runs) Decision: None (plant runs) Net: $15,000,000 Flexibility: Wait Flexibility: Commit plant Flexibility: (Plant or) Not NPV $9.504M PV(E(C 1 )) = $2.5/1.1 $2.273M PV(E(C 2 )) = $8.75/1.1 2 $7.231M FIGURE 12.5 Value to Flexible Plant Starting (But Not Stopping) shows the tree for this scenario the plant value now comes to +$9.5 million. This is more than you get from the option to delay in this scenario, because there is one node (where the price hits $600/unit) where you now could make money where previously you had to have already committed yourself not to operate. (The relevant box that is different is the one with the red box.) But this is less than what you get under perfect flexibility, because you are still robbed of the option to shut down if the retail price is $400/unit in the final period. The option to stop later: Yet another alternative scenario would force you to keep a once-closed plant stopped. That is, you cannot restart a plant once you have

12.10 DECISION TREES: ONE SET OF PARAMETERS 439 Flexibility: Try plant Flexibility: Abandon plant NPV $5.372M Flexibility: Abandon plant PV(E(C 1 )) = $2.5/1.1 $2.273M Decision: None (plant is dead) Decision: None (plant is dead) PV(E(C 2 )) = 3.75/1.1 2 $3.099M FIGURE 12.6 Value to Flexible Plant Stopping (But Not Starting) Strategy 1: Close at $400 shut the burners off and allowed your skilled workers to leave. This is called the abandonment option. This case also illustrates that decision trees can become complex. If the price falls to $400/unit at first, should you run the plant or not? If you do not run the plant, you save money but you lose the real option to operate if the price then appreciates to $600/unit. Actually, you have no choice but to compute the best value both ways. Figure 12.6 and Figure 12.7 show the two decision trees. If you close the plant, your firm would be worth $5.4 million (Figure 12.6). If you keep the plant open eating a loss of $15 million rather than just $10 million that first year your firm would be worth $8.3M, because you keep the real option to operate if the retail price were to increase again to $600/unit. Therefore, keeping the plant open is the better strategy. What should you do if the price falls to $400/unit at year 1?

440 CHAPTER 12 APPENDIX VALUING SOME MORE REAL OPTIONS Flexibility: Try plant Flexibility: Abandon plant Flexibility: Keep plant alive Net: $15,000,000 Flexibility: Abandon plant NPV $8.264M PV(E(C 1 )) = $0M PV(E(C 2 )) = $10/1.1 2 $8.264M FIGURE 12.7 Value to Flexible Plant Stopping (But Not Starting) Strategy 2: Run at $400 You really need to consider all possible future strategies in response to all possible future price paths. Solving such trees is a difficult problem, because your optimal strategy next year does not just depend on that year but also on future years. In fact, in our previous examples, I have cheated in making it easy for you: I had told you the strategy at each node. Real option problems are difficult to value, precisely because your optimal strategy at any node can depend both on the current state of your firm and on all future possible scenarios. The web chapter on real options explains how you can solve such problems more systematically. Decisions are often worked out backwards : You start with the final year and work your way toward today. Another important tool is the

12.11 PROJECTS WITH DIFFERENT PARAMETERS 441 aforementioned scenario analysis, which simply means trying out different input values some more pessimistic to see how they impact the estimated value of a project. (Scenario analysis and sensitivity analysis are very similar. The former is sometimes used as the name if more than one input value is changed; the latter if only one input value is changed.) Finally, also explained in the web chapter, there is a form of automated scenario analysis (called Monte Carlo simulation), in which you can specify a whole range of possible future scenarios. The spreadsheet itself can then compute the expected outcomes in many different scenarios using different decision-making strategies that you would specify. 12.11 PROJECTS WITH DIFFERENT PARAMETERS This example was a little artificial, because it kept the same parameters throughout. This symmetry made it easy to explain and compare options. More commonly, the parameters themselves will change and determine the extent of your flexibility (and thus the value of your real option). This is best explained by example. Consider how, in the real world, different projects have different parameters. Different projects are different bundles of real options. Fixed versus flexible technology choice: Let s assume that you have a factory with a fully flexible technology, as illustrated in Figure 12.3. I am now offering you an alternative technology, which eliminates your fixed operating costs of $50 million per year but requires a one-time upfront $80 million investment. (You are installing robots that will replace expensive manpower.) At first blush, this seems like a great idea you no longer have to spend $100 million, which discounts to $50/1.1 + $50/1.1 2 $86.777 million today. But is this really a savings of $6.777 million for you? No. It ignores the real option of flexibility that human workers have over robots: They can be hired and fired. Once purchased, robots cannot be laid off depending on demand. Figure 12.8 shows that with the robots you would have, you end up with $6.777 million, rather than $10.537 million. Robots, therefore, are not a great idea. Incidentally, it is often suggested that the value of smart employees is not their initial or even expected value, but the fact that smart people have the flexibility to attack novel problems for which they are not initially hired. Think about it your value may be primarily that of a real option! Adding plant capacity: Another interesting real option is the option to expand. You can view this as the choice to build currently unused capacity. For example, say you can choose between two options: Your current fully flexible production technology that allows you to produce 150,000 units at $100/unit (as in Figure 12.3). Another production technology that builds the following extra capacity: You can still produce 150,000 units at $100/unit, but you can also double your production with 300,000 units at a cost of $200/unit, though with higher machine costs of $100,000. Note that doubling increases the cost of all goods, not just the cost of the extra 150,000 units. It would cost you $60 million in variable production costs rather

442 CHAPTER 12 APPENDIX VALUING SOME MORE REAL OPTIONS Flexibility: Technology? Fixed costs: $80,000,000 Net: $65,000,000 Net: $35,000,000 Net: $95,000,000 Net: $35,000,000 Net: $65,000,000 Revenues: $15,000,000 Net: $5,000,000 PV $86.777M NPV $6.777M PV(E(C 1 )) = $50/1.1 $45.454M PV(E(C 2 )) = $50/1.1 2 $41.322M FIGURE 12.8 Value of a One-Time $80 Million Fixed-Cost Technology with Different Parameters (no more fixed costs per period, but a one-time upfront expense) than just $15 million, and $100 million in fixed costs rather than just $50 million that is, almost $95 million more if you ever wanted to use such extra capacity! Would you be willing to pay $3 million to upgrade your plant to such a technology? Figure 12.9 shows you the firm value with the option to expand. If the retail price hits its all-time high of $800/unit, the unused capacity is worth a tremendous amount. Therefore, the value of the firm increases to $15.7 million from your earlier optimal value of $10.5 million, easily enough to justify a $3 million expenditure. solve now! Q 12.44 A business produces 100,000 gadgets that cost $1 each to produce and sell for $1.80 each (last year and just now). To produce another 100,000

12.11 PROJECTS WITH DIFFERENT PARAMETERS 443 Retail P = $800, Cost C = $200 double Revenues: $180,000,000 Fixed costs: $100,000,000 Net: $70,000,000 Flexibility: Build capacity? Fixed costs: $3,000,000 PV $15.703M NPV $12.703M PV(E(C 1 )) $2.5/1.1 $2.273M PV(E(C 2 )) $16.25/1.1 2 $13.430M FIGURE 12.9 Value of an Expansion Technology with Different Parameters (relative to Figure 12.3) gadgets requires running the machine at night, which increases production costs from $1 to $2. The business can last for up to 2 years (but think about how you would solve this for 5 years). In every year, with 10% probability, the output price doubles; with 10% probability, the output price halves; and with 80% probability, the price stays the same as in the previous year. Shutting down the factory for 1 year costs $9,000. Reopening it costs $10,000. The cost of capital is a constant 5% per year. What is the value of this factory? (This is a difficult problem, but unfortunately not an unrealistic one.)