This chapter offers a much more detailed discussion of real options compared to what youlearned originally in Chapter ??. (You must read Chapter ?? before you read this web chapter!)Assessing the value of real options is as important as it is difficult. This web chapter shouldhelp you learn how to think about and solve these difficult NPV problems. It relies more heavilyon statistical concepts than the rest of the book.
Side Note: Some academics refer exclusively to “valuation by replication” (explained in Section 3·3.D) as the“real options” approach. However, this chapter refers to “real options” as options embedded in real projects,which can be valued through a number of different techniques, “valuation by replication” being one of them.
Anecdote: An Early Real OptionThe earliest known option contract was also a real option. It was recorded by Aristotle in the story of Thales theMilesian, an ancient Greek philosopher. Believing that the upcoming olive harvest would be especially bountiful,Thales entered into agreements with the owners of all the olive oil presses in the region. In exchange for a smalldeposit months ahead of the harvest, Thales obtained the right to lease the presses at market prices during theharvest. As it turned out, Thales was correct about the harvest, demand for oil presses boomed, and he madea great deal of money.
The word option is a synonym for choice—the ability to do something or not do somethingExplicit and ImplicitFinancial Options. in the future, at your discretion. Financial options are common. The most familiar are stock
options, which are traded, e.g., on the Chicago Board Options Exchange (CBOE). The prices ofthese options can be looked up on Yahoo!Finance. Some options are traded over the counter:if you want to purchase a 10-year option on the Sony, chances are that you would have to asksomeone—typically an investment bank—to manufacture such an option for you. Other finan-cial options are embedded in contracts and securities. For example, your mortgage contractmore than likely gives you the option to pay off the mortgage at your discretion, which youshould do (and refinance) if interest rates drop enough. Your car insurance liability may have adeductible, which de facto means that the insurance is only an option that gives you the rightto exercise it if the damage exceeds the deductible.
But this chapter is not so much interested in financial options as it is in real options. What is theReal Options: Yourchoice to alter
operations in the future.difference? A real option differs from a financial option in that the exercise of the real optionrequires a change in the physical, “real” project. Such real option projects can be factories,buildings, R&D activities, and so on. The most prominent real options are
Timing Your ability to start or stop a project at a time of your discretion.
Abandoning Your ability to abandon a project at a time of your discretion.
Accelerating Your ability to speed up a project at a time of your discretion.
Expansion Your ability to expand a project at a time of your discretion.
Switching Your ability to switch to a different technology.
Real options are difficult to value, but no one would argue that this means that you can ignorethem. In fact, valuing real options is often more important than getting the discount rate right.You have no alternative but to give it your best shot.
Anecdote: Real or Unreal?In their article on real options in the Harvard Business Review (March 2004), Copeland and Tufano cite aBain&Company “Management Tools and Techniques” 2001 survey of 451 senior executives who had tried thereal-options approach. One-third had given up on it the very same year—and I would guess that most of thesemanagers had only adopted the simplest real option techniques, to begin with.
Whose fault is this? Is it primarily the fault of the technique? No! It is primarily the fault of the problems.Projects with real options are usually very difficult to value, and the only alternative is the “head-in-the-sand”method. Neither does this ostrich method work better, nor is ignoring real option values often just a modesterror in terms of the value. It might be first-order!
The best way to understand real options is to work an example. Say it is the end of 1999, and you The scenario inputs.
are considering purchasing a lease for a combined cycle gas-turbine electricity generation plantwith 400MW capacity. Your input, natural gas, has a fairly constant price of $7 per thousandcubic feet, so your plant’s total cost of electricity production is $25/MWh. Table III.1 showsthe electricity prices for 1999. What is the per-week value of leasing this plant?
Table III.1. Average Weekly Wholesale Price of Electricity Per MWh at the California PowerExchange in 1999.
Jan 1 - Jan 2 $$12.89 May 2 - May 8 $$23.46 Sep 5 - Sep 11 $$28.61May 9 - May 15 $$22.68 Sep 19 - Sep 25 $$31.91
Jan 3 - Jan 9 $$23.14 May 16 - May 22 $$26.08 Sep 26 - Oct 2 $$49.71Jan 10 - Jan 16 $$24.13 May 23 - May 29 $$26.63 Quarterly Average $31.90Jan 17 - Jan 23 $$20.78 May 30 - Jun 5 $$15.82Jan 24 - Jan 30 $$18.29 Jun 6 - Jun 12 $$16.49 Oct 3 - Oct 9 $$43.25Jan 31 - Feb 6 $$18.89 Jun 13 - Jun 19 $$25.29 Oct 10 - Oct 16 $$52.59Feb 7 - Feb 13 $$19.57 Jun 20 - Jun 26 $$26.30 Oct 17 - Oct 23 $$45.34Feb 14 - Feb 20 $$18.78 Jun 27 - Jul 3 $$32.26 Oct 24 - Oct 30 $$44.10Feb 21 - Feb 27 $$18.95 Jul 4 - Jul 10 $$23.13 Oct 31 - Nov 6 $$59.84Feb 28 - Mar 6 $$17.70 Quarterly Average $24.02 Nov 7 - Nov 13 $$40.15Mar 7 - Mar 13 $$17.79 Nov 14 - Nov 20 $$30.44Mar 14 - Mar 20 $$19.76 Jul 11 - Jul 17 $$40.80 Nov 21 - Nov 27 $$30.40Mar 21 - Mar 27 $$18.31 Jul 18 - Jul 24 $$24.31 Nov 28 - Dec 4 $$28.25Mar 28 - Apr 3 $$20.15 Jul 25 - Jul 31 $$26.58 Dec 5 - Dec 11 $$30.03
Quarterly Average $19.71 Aug 1 - Aug 7 $$28.19 Dec 12 - Dec 18 $$31.26Aug 8 - Aug 14 $$23.05 Dec 19 - Dec 25 $$28.31
Apr 4 - Apr 10 $$23.31 Aug 15 - Aug 21 $$26.59 Dec 26 - Dec 31 $$28.89Apr 11 - Apr 17 $$25.23 Aug 22 - Aug 28 $$47.26 Quarterly Average $37.90Apr 18 - Apr 24 $$26.83 Aug 29 - Sep 4 $$33.64Apr 25 - May 1 $$21.88 Sep 12 - Sep 18 $$30.95
Mean $28.09 Median $26.30Standard Deviation $10.11Range: $12.89–$59.84Interquartile Range $20.78–$30.95
3·2.B. Recognizing The Real Option
Historically, according to Table III.1, the average electricity price was around $28/MWh, and The value in the mostlikely scenario.this is not far off from the current electricity price of $28.89 (standing at the end of 1999). So, a
rough naïve estimate would be that with 400MW capacity and $3.09 profit per hour and for the8,765.8 hours per year, you would have earned a net profit of 400MW·$3.09/MWh·8,765.8h ≈$10.8 million in 1999.
But this naïve valuation would have been wildly incorrect. The reason is that you have a real The value is much higherif you recognize thatyou can shut down andreopen.
option: you can shut down your gas turbines when the price of electricity is low. The valueof your option depends on how quickly you can shut down and reopen your plant—and howmuch doing so would cost. If you simplify the problem, assuming that you can only shut downonce a week and without cost (conservative), but that you know what the average price duringthe entire week will be (aggressive), you can determine the profits you could have made in 1999.There were 30 weeks in 1999 during which the price of electricity exceeded $25. The averageprice in these weeks was $34.19. If you had operated only during these weeks, you would haveearned $34.19− $25 ≈ $14.19/MWh, albeit only for 30 · 24 · 7 = 5,040 hours. Your cash flowwould have thus been $14.19/MWh·5,040h·400MW ≈ $28.6 million—almost three times what
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the naïve valuation had suggested. Ignoring the real option to shut down and reopen wouldnot have been a forgivable valuation mistake!
3·2.C. Variance, Option Value, and Technological Choice
An important insight is that if you own a real option, variance helps you: If the price had beenVariance increases thevalue of flexibility. $28.09/MWh constant, you would have earned $10.8 million and you would never have shut
down or reopened the plant. But because the price was highly variable around $28.09/MWh,you would have earned $28.6 million. It is your ability to “operate only when desired” that hasvalue. Intuitively, “the bigger the upside,” the better for you. “The lower the downside” makesno difference: you are not operating anyway. Variability is on your side!
Important: Real options are more valuable if there is more variance.
This does not contradict our intuition from the investments section. There, we posited that youdisliked risk (at least systematic risk), and would only take it on if you receive extra expectedrate of return. You may still intrinsically dislike risk, and require a higher hurdle (discount)rate for investments with much embedded real option values—such cash flows are typicallyvery risky. But, in the presence of a real option, the risk also increases your expected cashflows—and often so tremendously that you end up much better off with risk than without risk.The present value, taking the higher discount rate into account, can be much greater.
Real options generally arise from your flexibility (here, whether to operate or not to operate).Fixed cost technologieshave fewer real options. Different technologies have different real options and to different extents. For example, nuclear
power plants have higher upfront fixed costs, but lower marginal costs. A nuclear power plantmay cost over $1 billion to construct, but it may be capable of producing electricity at costsas low as $5/MWh. Therefore, nuclear energy plants typically run continuously, regardless ofelectricity price. They have fewer embedded real options. This has another consequence: If youignore real options, you will mistakenly end up with too many high fixed-cost, low variable-costtechnologies. You will believe nuclear power plants are much better than turbine gas plants,even if they are not.
Indeed, many economic resources are nothing but real options. Much R&D, e.g., into a newMany assets areprimarily real options. cancer drug, will never pay off in and of itself. But, if the drug development were to succeed,
the pharmaceutical company would create factories and earn billions of dollars. An investmentof R&D can thus be considered the purchase of a real option. Similarly, undeveloped land haszero inflows today, and still requires the payment of real estate taxes. Its only value is the realoption to build on it if demand for land use were to increase in the future. And, your degreeand next job may have more value, because you can walk away from them if other, betteropportunities were to appear.
Solve Now!
Q III.1 Is it possible that you may prefer a scenario in which in the most likely scenario, youwould do worse?
Q III.2 Compare two technologies. The first can produce electricity at $28/MWh, but cannotclose down and reopen. The second can produce electricity at $29/MWh, but can close down andreopen. In the most likely price scenario, which technology would have earned more? Whichtechnology is worth more?
Q III.3 Do variable-cost technologies or fixed-cost technologies usually contain more real optionvalue?
Section 3·2. Valuing A Gas Turbine Plant.
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3·2.D. Real-World Complications
By recognizing your ability to shut down the gas turbine plant, you are able to estimate a much Problems in predictingcash flows.more precise value than you would assuming a stupid “always running” plant value. However,
many problems remain with the $28.6 million value estimate. Even if you cannot address eachand every one, you want to at least recognize them.
Your most important concern is that you need to estimate the electricity price process for 2000, Electricity price processpredictionnot for 1999. This is a self-contained task—you can do so, even before you ever consider the
plant itself.
1. Your $28.6 million calculation was for profits if you had operated in 1999. Alas, it is The future is not exactlythe past.more sensible to assume that 1999 would only be indicative of 2000 in terms of the
electricity price process (e.g., mean and standard deviation). Moreover, you know that1999 closed with a price of $28.89, not the average price of $28.09. Can you make useof this knowledge? In any case, you should assume that history will not repeat itself,but that it can teach you a lot about the future. This is exactly analogous to our earlierassumption in the investments part of our book, where we stipulated that historical returnstatistics (means and standard deviations) are indicative of future statistics, but that theexact future return realization could end up quite different from the past.
2. Even if you use 1999 data to estimate the statistical process for 2000, it could be that The future may not evenbe indicated by the past.the historical process is not a good guide for your future. If your experience in electricity
management makes you capable of better estimating the price of electricity in 2000, thenyou can come up with a better cash flow estimate. But to improve your estimate of thevalue of the plant, you must be able not only to forecast the mean electricity price, butalso the standard deviation in the electricity price. (In retrospect, we know that the pastturned out not to be indicative of the future. California experienced a famous energycrisis in 2000, partially due to price manipulation by the energy producers themselves!)
3. If you look at the historical prices, you will see that each weekly price is not exactly You need a goodprediction for electricityprices, standing at$28.89.
a random draw from a fixed distribution. When the price the previous week was high,chances are that the price this week is also high. For example, at the end of the year, theprice stood at $28.89. This implies that the next price is unlikely to be $50 or $18, butmore likely something between $26 and $32. In contrast, when the price was $50 lastweek, you would expect it to be something, say, between $44 and $52. (Note that this isnot necessarily symmetric around $50!)
Side Note: For intuition, think of this process (in which recent prices are indicative of the next price)as being similar to, but not the same as the random-walk process. The reason is that there need to bereasonable limits to how far the electricity can wander off before demand drops and supply increases.Statisticians call such processes “autoregressive,” and we will discuss this in much detail below.
Note also that there could be some seasonality: electricity prices in summer could be higher, but youwould need more years of historical data to judge whether this is the case. Thus, we ignore seasonality.
You know that in order to value your plant, you must forecast some process for the electricity Prediction of theElectricity Time-SeriesProcess can be done inmany different ways.
price in 2000. Your technique itself can range from the simple to the fancy. In the real-world,the better you are at electricity price process estimation, the better will be your value estimates.Typically, the best forecasts combine qualitative managerial knowledge of the industry andbusiness with quantitative statistical modeling techniques. We shall explore two kinds of elec-tricity forecasting processes in the next two sections: a simple qualitative version, in which youwrite down a reasonable and intuitive binomial tree (Section 3·3); and a quantitative statisticaltime-series estimation, in which you estimate a time-series model from the historical electricityprice data (in Section 3·4).
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Of course, your generation plant introduces its own set of difficulties, too:Electricity Price ProcessPrediction
4. It is not clear that one week is an appropriate time interval for your analysis. In the back-You may needappropriate (shorter)
time intervals.of-the-envelope valuation, we simply assumed that at the beginning of each week, youwould know the fixed average electricity price for this particular week—but in real life,electricity prices may fluctuate by the minute!
5. Even if electricity prices vary by the minute, it is not clear if your technology allows youYou need to know thelimits of your
technology.to react instantly. Can you shut down your gas turbines the minute the price falls below$25/MWh, or does it take several hours? Can you restart them immediately? If youcannot act instantly, then you do not need to know the minute-by-minute electricity priceprocess. If you can act instantly, you might have to perform your electricity price analysisover much shorter frequencies than weekly. If you can act only very slowly, maybe youshould just forecast your electricity price process over, say, quarterly horizons.
6. In the example, your only choice is whether to operate or not to operate. What if youYou need to knowwhether you have the
option to “work harder.”can produce energy at various prices in various plants? For example, you may have somenewer turbines that can produce at $20/MWh and some older ones that can produce at$30/MWh. Now, you have to make multiple decisions: when to switch on/off the firstturbine, and when to switch on/off the second turbine. Indeed, in many situations, theproduction cost function may be smooth over some range: if you drive the machine andemployees harder, you can produce more electricity, but at a higher per-unit generationcost.
And, do not forget about the market, both the electricity and the financial markets:
7. Can electricity always be sold, or do you need to search for consumers? If you are a largeYou need to knowwhether all electricity is
always sold.enough power producer, your power generation could itself have a nontrivial effect onthe electricity market price!
8. Electricity is not your only source of uncertainty. For example, the price of natural gasYou need to know whatother important factors
are uncertain.could also be uncertain—and it could correlate with the price of electricity. This matters: ifthe price of your input, natural gas, always goes down when the electricity price goes down,you may always choose to operate. However, if the price of your input varies inverselywith your output price (gas would be cheap when electricity is expensive), then you maybe in an even more volatile situation—where your real option is even more valuable.
9. What is the appropriate discount factor for discounting cash flows? How do your cashDiscount Factor?
flows covary with the rate of return on the stock market? Of course, you know that fora 1-year project, the net present value is not particularly sensitive to reasonable errorsin your discount rate estimate, so this is probably not an issue of first-order importancehere. But, what if you were to consider leasing the plant for many years? Then estimatingthe appropriate discount rate would again become an issue of great importance. (In somespecial situation, you may not even need to estimate the discount rate, but can derive avalue purely by replication, as explained below.)
Section 3·2. Valuing A Gas Turbine Plant.
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3·2.E. Path (History) Dependence
These are all difficult questions—but your problem becomes exponentially more difficult if If you can restrict yourview to just the currenttime, the optimizationproblem becomesrelatively easy.
your best operating decision depends not only on the current electricity price, but also on whathappened earlier or what can happen later.
Dependence on history can happen if it is expensive to change the operating state. For example,If you have to pay tochange state (factoryopen or closed), thenyour choice depends onyour current state,which depends on yourhistory.
assume that it costs $200,000 to shut down your plant, and another $500,000 to restart it.Whether you operate or not then depends not only on the current price of electricity, but alsoon the state that your operations are in. For example, recall that your electricity productioncost is $25/MWh, and presume now that the current electricity price has just moved to $24.99.
• If the plant was already closed, the choice is easy: keep the plant closed. After all, re-opening would cost money today, plus you would lose money on each MWh you wouldproduce.
• If the plant was open, should you shut it down? Probably not—the shutdown and reopen-ing costs are so high, you would probably be better off just accepting the 1 cent/MWh lossinstead of incurring the shutdown costs now and possible reopening costs next period ifthe price were to go up again (of course, unless you would be at the end of your lease).
But where is the cutoff price below which you should shut down? What if the price were $24.85?Similarly, at what price should you reopen a closed plant? Should you reopen at $25.01? Again,the answer is probably no: the $500,000 cost of reopening today plus the reclosing costs if theprice were to drop again next week would be prohibitive.
This set of issues is called path dependence, because your choice depends not only on the The current statedepends on where wecame from.
current market situation (the electricity price), but also on the path you have taken—the stateof your plant, which in turn depends on whether the electricity price has just dropped a lot (inwhich case your plant would likely be open) or whether it has just increased a lot (in which caseyour plant would likely be closed). In other words, history matters. Of course, when historymatters, you also have to consider how your choice today becomes relevant history tomorrow.(In the real world, your problem can become even more complex, because your choice may alsodepend even more directly on the future: for example, it may be possible to store electricity. Ifso, you might still produce electricity if the current price were $22/MWh and hope for a higherprice later on. Of course, at some price low enough, you may not want to produce any moreelectricity.)
3·2.F. Preview
Now, one chapter cannot address all of these points—but it can explain the general techniques We pick and chooseissues to illustratesolution techniques.
of how you should approach them. The rest of this chapter shows you two different techniquesin how to deal with the most difficult problems:
• Section 3·3 focuses on how you can handle path-dependent decisions if you assume thatthe electricity price follows a “binomial tree” process. This method can often be executedon a sheet of paper with pencil and paper (and a calculator, of course) and lends itself touse by more qualitative and less quantitative managers.
• Section 3·4 models the price process for electricity (for subsequent handling of pathdependent decisions) in a more general “statistical time-series process” context. This canrequire a lot of statistical and computer expertise, but it is more adaptable and flexiblethan the binomial tree technique.
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3·3. The Simpler Method: Binomial Trees and Path Depen-dence
Our first method is “simple” (well, simpler), focusing more on getting the main value driversThis approach works forthe particular process
(binomial tree), andespecially if only a fewdecisions can be made.
right, instead of getting the electricity forecast details perfect. It posits a particular processfor the electricity price—a “binomial tree” (see below). It is typically used by managers forrelatively low-frequency processes, such as quarterly choices. Because it tends to work well fordecisions that have to be made only a few times, it might apply here if your electricity plantcan only be shut down infrequently (say, quarterly), perhaps because it requires a long time orbecause changing state costs a lot of money. (Without a computer program, this method wouldbe too tedious if you wanted to model daily decisions.)
3·3.A. Basic Processes and Decisions
To fit this description, assume that your plant is really less of a turbine gas plant and more of aDecisions are lessfrequent. nuclear power plant: you make only a once-a-quarter decision whether to operate or not operate
it. Because a standard year has 8,765.81277 hours, if you operate in one quarter, you will run theplant for all (approx.) 2,191.5 hours of the quarter, and earn a profit of 400MW · (Price− $25)per hour. For example, if the electricity price were $30/MWh, then you would earn in oneoperating quarter
(M$denotes million of dollars.) Your ultimate goal is to work out the value if you can lease theplant for this plus another four quarters.
Your first step is the most difficult, and the most critical—and the one where this section takesToughest part: estimatethe price process. its main shortcut: we assume a simple binomial tree process for how the price of electricity
can evolve. In the real world, its construction can rely not only on historical data, but also onyour managerial judgment. Your hope must be that this five-quarter binomial tree posits anelectricity process that captures the basic electricity price dynamics, rather than a process thatis the very best in heavy-duty statistical artillery.
As a good qualitatively-oriented manager, you should start with your intuition based on his-Our specific electricityprocess. torical experience. The electricity price in 1999 started at $23.14/MWh. If you compute the
average quarterly price in 1999, you find that it was $19.71/MWh in Q1, $24.02/MWh in Q2,$31.90/MWh in Q3, and $37.90/MWh in Q4. This means that, given the starting price, thepercent change to the first quarter average price was –15%. The remaining changes were +22%,then +32%, and +18%. As an experienced and knowledgeable electrical plant manager with ac-cess to this and other information, let us assume that you believe the future electricity price tobe well represented by a process that either increases by 20% in one quarter, or decreases by15%, give or take, and with equal probability. Thus, Figure III.1 represents your tree for possi-ble electricity prices. (Warning: All subsequent price, revenue, and present value calculationswill carry full precision internally. Be prepared for small rounding errors if you replicate thecalculations from the displayed prices and revenues.)
Section 3·3. The Simpler Method: Binomial Trees and Path Dependence.
Each path is equally likely, hence the probability of 1/2 on each arrow. An up-path is a 20% electricity price increase,a down-path is a 15% decrease. In the real world, you would round prices, but to make computations easier to checkfor you, prices have three digits after the decimal point.
Side Note: If you believe that the price cannot wander off too far, e.g. from $50/MWh, then you can changethe price on a node to something you consider to be more realistic. For example, you might change the $49.922price to $45, and the $59.906 price to $52. Use your managerial intuition! All subsequent methods will workwith a tree modified in this manner.
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3·3.B. Costly vs. Free Operating Changes
Your first benchmark is the value of the plant if you always operate it. With the electricityBenchmarks: No realoption. “Free exercise”
real option.price process in Figure III.1, and assuming that the discount rate is 2.5% per quarter, you wouldestimate your plant’s value V to be
(III.2)You already know that this ignores the real option to close the plant if the price of electricitygoes down. Your second benchmark is the value if you can shut down and reopen at no cost. To
Section 3·3. The Simpler Method: Binomial Trees and Path Dependence.
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compute this, simply replace 2,191.5h with 0h if the electricity price is below your generationcost of $25, and the plant’s value would be
We now introduce the complication which is the whole point of this section: shutting down Action changes arecostly.and reopening are not free. Let’s assume that laying off employees and mothballing the plant
will cost you $200,000. Rehiring them and restarting the plant will cost you another $500,000.Without closing and opening costs, you really had only two decisions—open the plant or shutthe plant. With opening and closing costs, you now have four potential decisions to consider:
If OpenKeep Open (KO)
Re-Close (C)��@R
If ClosedKeep Closed (KC)
Re-Open (O)��@R
Without the extra costs, the “keep open” and “open up” decisions were always the same, as werethe “keep closed” and “close down.” They did not depend on whether you came into a periodwith a closed or open plant—you would just do whatever would be best. Note that you canalso think of the value of the plant if always operating, and the value of the plant if perfectlyflexible as special cases of particular closing/reopening prices: If the closing cost were infinite(or at least very, very high), you would always operate, and thus obtain a value of M$23.541.If both opening and closing costs were zero, you would do whatever is currently optimal andthus obtain a value of M$26.792. Therefore, you already know that the true value of your plantmust lie between your two benchmarks.
Even with these costs, many decisions are still immediately obvious. If the electricity price is For most nodes,switching costs are toosmall too matter oneway or the other.
far away from your cost of $25, e.g., if electricity is $30, you earn so much money in one quarter(400MW · ($30/MWh− $25/MWh) · 2,191.5h ≈ M$4.383) that even if you were to arrive in thisquarter with a closed plant, you would still gladly incur a $500,000 restartup cost (and potentialfuture reclosing cost of $200,000). Similarly, if the electricity price were $20, operating the plantwould lose you so much money (400MW · ($20/MWh−$25/MWh) ·2,191.5h ≈ −M$4.383) thatyou would always shut down even if you were to come into this quarter with an open plant.
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Figure III.2. Obvious and Non-Obvious Decisions
Now Q1 Q2 Q3 Q4
P=$28.89⇒ Operate
(R = +M$3.410)
P=$24.556⇒ What to do?
(R ≈ −M$0.389)
P=$34.668⇒ Operate
(R = +M$8.475)
P=$20.873⇒ Don’t Op.
(R = −M$3.618m $0)
P=$29.468⇒ Operate
(R = +M$3.916)
P=$41.602⇒ Operate
(R = +M$14.553)
P=$17.742⇒ Don’t Op.
(R = −M$6.362m $0)
P=$25.048⇒ What to do?
(R ≈ +M$0.042)
P=$35.361⇒ Operate
(R = +M$9.083)
P=$49.922⇒ Operate
(R = +M$21.847)
P=$15.081⇒ Don’t Op.
(R = −M$8.695m $0)
P=$21.290⇒ Don’t Op.
(R = −M$3.252m $0)
P=$30.057⇒ Operate
(R = +M$4.433)
P=$42.434⇒ Operate
(R = +M$15.282)
P=$59.906⇒ Operate
(R = +M$30.599)
R is the single-period revenue of the plant if open and operating. Opening and closing costs are ignored. The boxednodes are where your decision is not obvious, and requires more thought. Revenues are in million dollars (M$).
Section 3·3. The Simpler Method: Binomial Trees and Path Dependence.
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Can you determine some price ranges where you do not need to bother with much thinking? Here are someboundaries as to wherewe have obviousdecisions and where not.
Yes! Pick the total reopening and shutting down costs of $700,000. If by operating, you wereto earn more than $700,000, you would clearly always start up, even if you were coming inclosed. Similarly, if by shutting down, you were to avoid a loss of more than $700,000, youwould always shut down, even if you were operating. You can determine the price at which thishappens by solving
R( P ) = 400MW · (P − $25/MWh) · 2,191.5h = −$700,000 ⇒ P ≈ $24.20 .(III.4)
So, you do not have to do any thinking if the electricity price is above $25.80 or below $24.20.You will always open (or shut) the plant. It is only for electricity prices in between $24.20and $25.80 that you shall have to turn on your brain. Your knowledge so far is illustrated inFigure III.2. For later convenience, I have also already added the value of R(P ) if you were tooperate at each node.
Although your logic has been impeccable, this is not a systematic way to approach this problem— We have not solved outproblem!it will not help you determine what you should do if the price is between $24.20 and $25.80.
For example, what should you do if the price were $24.556 in Q1 and you came in with anopen plant? Or, if the price were $25.048 in Q3, and you came in with a closed plant? You willtherefore have to learn how to work out your problems in a more systematic way.
3·3.C. Solution Techniques for Dynamic Problems
The Tree Setup
So how can you work these kinds of problems more systematically? How can you handle tree The two cardinal rules:work backwards, andconsider decisionscontingent on state.
nodes where decisions are not obvious? You need an algorithm that tells you how to approachthe problem systematically. There are two main rules you need to follow.
1. Work backwards.
2. Keep track of both best choices (decisions) and the current state when working the deci-sion tree.
It will soon become clear exactly what these two rules mean. Figure III.2 provides your basictree. (Figure III.2 only carries the operating revenues, ignoring any possible reopening costs.)At each node, you have two choices—what you would do if you come in with an open plant, andwhat you would do if you come in with a closed plant. If you arrive with an open plant, yourchoices are to “keep open” or to “close down.” If you arrive with a closed plant, your choices areto “keep closed” or to “open up.” The revenues of your plant and thus the plant value dependson your action, and must also be carried in the diagram. Plus, if you determine that you wantto shut down the plant at one particular node, you have to remember that you will arrive inthe next quarter with a closed down plant, which will itself have value implications. Let us nowwork out the details of the valuation.
You already know the correct answer for many of the nodes (from Formula III.4 on Page 65), OK, we will take theextra slow route now.which could make your life easier. But we want to develop a method to work problems of this
kind systematically, rather than piecemeal. So, please bear with my more tedious method fornow.
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Quarter 4
Your first rule is to always work backwards, so you start with the nodes at the right end of theThe decisions in the finalperiod are worked out
first.tree for Quarter 4. The big advantage of starting at the end is that nothing comes thereafter—soyou do not have to worry about what impact your Q4 decision may have the following quarter.Now, you need to fill in what the value would be, depending on the current plant state, for eachnode. Thus, you need to determine the best action (with consequent value) contingent on aplant that is “arriving while open” and that is “arriving while closed.” You already know thecorrect decisions. For example, if the price is $59.906, you should operate, regardless of priorstate. So here is the complete list:
P = $59.906, arrive open: The price is so high, you want to be open. Recall that the quarterlyrevenue function is
The plant net revenue is just the electricity produced, V = R(P ) = +M$30.599.
P = $59.906, arrive closed: The revenues would be M$30.599 minus the reopening costs fora net of M$30.099, but this is irrelevant. If you think about this node, reflection will tellyou that you can never come to this node with a closed plant. That is, in order to reachthis node, you would have to have experienced three up moves—in which case the pricewould have been $49.922, and your plant would already be open.
P = $42.434, arrive open: You always operate, so the plant net revenue is +M$15.282.
P = $42.434, arrive closed: Again, this cannot happen. Therefore, you can ignore this possi-bility.
P = $30.057, arrive open: The net revenue is +M$4.433.
P = $30.057, arrive closed: This can happen, because the price may have just risen from$25.048! The revenue is +M$4.433 minus the reopening costs of $500,000 for a net of+M$3.933.
P = $21.290, arrive open: You would not want to operate, so the net revenue is $0, minusclosing costs of $200,000 for a net of -M$0.2.
P = $21.290, arrive closed: The value is $0.
P = $15.081, arrive open: Never happens. If it did, you would pay $200,000 to shut down,for a net of -M$0.2.
P = $15.081, arrive closed: The value is $0.
Now, write these best actions and resulting values in Q4 into your tree to arrive at Figure III.3.
Section 3·3. The Simpler Method: Binomial Trees and Path Dependence.
V means “value,” which includes both current revenue and discounted expected future revenues, assuming optimalfuture behavior, as well as opening and closing costs. One period revenue, if operating, is R(P ) = 400MW · (P −$25/MWh) · 2,191.5h. Present values (V ) include subsequent events, and are quoted in millions of dollars (M$). Ifyou arrive with an open plant, “Keep Opn” means your best action is to keep the plant open, “ Reclose” means yourbest action is to shut down the plant. If you arrive with a closed plant, “Keep Cld” means your best action is to keepthe plant closed, “ Reopen” means your best action is to open up the plant.
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Quarter 3
Now continue to work backwards. In Q3, your best courses of actions are obvious if the priceThe values at the threenodes where you know“whether to operate or
not” are easy tocompute.
is $49.922(operate), $35.361(operate), or $17.742(do not operate). You can quickly work outthe value in the tree if you are standing at any of these nodes. Start with the node where theprice is $49.922. You would earn current revenues R(P = $49.922 ) = +M$21.847 in Q3. Inaddition, looking at Figure III.3, you see that you will then either earn R($42.434 ) = +M$15.282or R($59.906 ) = +M$30.599 in Q4, to which you have to apply your discount rate of 2.5%.Therefore,
P = $49.922 and arrive open: V = +M$21.847 +1/2 · +M$15.282+ 1/2 · +M$21.847
1+ 2.5%= M$44.228 .
value today + disc exp value next quarter
(III.6)This V is the present value not just of the current-period (Q3) revenues at this node, but thetotal value of the plant including all future net revenues. Although it cannot happen, if youcame in with a closed plant, you would immediately open it up and earn
P = $49.922 and arrive closed: V = M$44.228− M$0.5 = M$43.728 . (III.7)
You can easily work the same calculation for the other two obvious prices, $35.361 and $17.742.The present value at these nodes are
P = $35.361 and arrive open: V = +M$9.083 +1/2 · +M$4.433+ 1/2 · +M$15.282
1+ 2.5%= M$18.700
P = $17.742 and arrive closed: V = $0 +1/2 · $0+ 1/2 · $0
1+ 2.5%= $0
value today + disc exp value next quarter .
(III.8)Each of these two nodes has two other possible states, which are just the preceding valuesminus the opening and closing cost, respectively.
P = $35.361 and arrive closed: V = M$18.700 − M$0.500 = M$18.200
P = $17.742 and arrive open: V = −M$0.200 + $0 = −M$0.200 .(III.9)
What you should do if you arrive at P = $25.048 in Q3 and you are closed? The answer is easyEven the one tough nodeis easy, if you arrive with
the plant open.if you arrive open: the plant is earning money on the margin, so you may as well just keep theplant open. An open plant earns
This is because next quarter, if the price goes up to $30.057, you continue operating andearn +M$4.433; if the price goes down to $21.290, you close down, suffering a closing costof $200,000.
Section 3·3. The Simpler Method: Binomial Trees and Path Dependence.
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The real interesting choice, however, occurs if you arrive at the node where P = $25.048 in Q3 And here is the toughdecision: what to do ifyou arrive closed? Youneed to compare values.
and you are closed. What should you do? You have to make one of two choices:
Keep Plant Closed If you were keep the plant closed, you would earn nothing in Q3. In addition,you would arrive in Q4 with a closed plant. This means that you would keep the plantclosed in Q4 if the electricity price were to drop to $21.290, but you would open it andearn M$4.42 minus the $500,000 reopening costs next quarter if the electricity price wereto increase to $30.057. Therefore, the plant’s present value if you keep it closed todaywould be
P = $25.048 and arrive closed and you keep plant closed :
V = $0 +1/2 · $0+ 1/2 · (+M$4.433− M$0.5)
1+ 2.5%= M$1.919
value today + disc exp value next quarter .
(III.12)
(Note that the M$0.5 reopening costs would occur the quarter thereafter, not here in Q3.)
Open up Plant If instead you were to open up the plant, you would earn an immediate $43,680minus the reopening costs of M$0.5 in Q3. If next quarter the price of electricity were tofall again, then you would need to shut down the plant down again. Of course, if the pricewere to go up, you could then just leave the plant open. In sum,
P = $25.048 and arrive closed and you open up the plant :
Because M$1.919 is more than M$1.607, the better choice is to keep the plant closed. So, theoptimal strategy if the price is $25.048 is
If P = $25.048 in Q3, If you arrive open, keep open V = M$2.107
If P = $25.048 in Q3, If you arrive closed, keep closed V = M$1.919 .
Enter this information into your tree to arrive at Figure III.7.
You should actually do such comparisons—is the value higher if my action is to keep the plantin its present state or if my action is to switch the plant to the opposite state? You need towork through the comparisons at every node of the tree, and then choose the action with thehigher value. However, in our other nodes, the right choice was so obvious that we did not haveto go through the painful computations.
Digging Deeper: It is here, at the price of $25.048, that your tree has become path-dependent: your bestaction depends on whether the price has just increased (coming from $20.873, in which case you would be closedand not operate) or decreased (coming from $29.468, in which case you would be operating and continue tooperate). Without path dependence (here caused by opening and closing costs), you could just work out the treeat each node independently of any other node in the tree—a much easier task.
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Figure III.4. Tree With Quarters Q3 and Q4 Worked Out
V means “value,” which includes both current revenue and discounted expected future revenues, assuming optimalfuture behavior, as well as opening and closing costs. One period revenue, if operating, is R(P ) = 400MW · (P −$25/MWh) · 2,191.5h. Present values (V ) include subsequent events, and are quoted in millions of dollars (M$). Ifyou arrive with an open plant, “Keep Opn” means your best action is to keep the plant open, “ Reclose” means yourbest action is to shut down the plant. If you arrive with a closed plant, “Keep Cld” means your best action is to keepthe plant closed, “ Reopen” means your best action is to open up the plant.
Section 3·3. The Simpler Method: Binomial Trees and Path Dependence.
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Quarter 2
Again, you work backwards. Quarter Q2 is relatively easier, because all decisions are clear. One Now you get somethingto do, too!of the exercises below asks you to check these values in Figure ?? yourself.
Quarter 1
Again, you continue working backwards. In Q1, if the price is $34.668, then the plant value An obviousdecision—$34.668is toohigh a price to notoperate.
is easy to compute, because the correct decision is obvious—operate. If you arrive open, thevalue is M$8.475, plus the expected discounted value of M$44.250 and M$14.114,
V = M$8.475 +1/2 · M$45.249+ 1/2 · M$14.006
1+ 2.5%= M$37.409 .
value today + disc exp value next quarter .(III.14)
If you could arrived closed (which never happens), you would immediately reopen the plantand earn M$36.909. Enter these V ’s into Figure ??.
But Q1 has another, more interesting decision. If the price is $24.556, what should you do? Of Q1 has anotherinteresting choice.course, if you come in closed, you can just leave it closed, because you are not earning money
from producing electricity on the margin. (The plant’s present value of the future then is reallyall you have, which is
P = $24.556 and arrive closed and you keep it closed :
V = $0+1/2 · M$13.566+ 1/2 · M$0.936
1+ 2.5%= M$7.074 .
(III.15)
However, if you come in open, should you close the plant? This is again a more difficult decision,and requires a full calculation:
Close Down Plant: If you close the plant, you have to pay $200,000 in shutdown costs, you willcome into the next quarter with a closed plant, which means that its value will be eitherM$13.566 or M$0.936 next quarter. Thus,
P = $24.556 and arrive open and you close down the plant :
V = −M$0.2 +1/2 · M$13.566+ 1/2 · M$0.936
1+ 2.5%= M$6.874
value today + disc exp value next quarter .
(III.16)
Keep Plant Open: If you keep the plant open, you lose ($25 − $24.556) = $0.444 per MWhproduced, for an immediate operating loss of M$0.389. In addition, you know that youwill come into the next quarter with an open plant, and that the value of this plant will beeither M$14.066 or M$0.736. Therefore
P = $24.556 and arrive open and you keep the plant open :
V = −M$0.389 +1/2 · M$14.066+ 1/2 · M$0.736
1+ 2.5%= M$6.832
value today + disc exp value next quarter .
(III.17)
Therefore, it appears that if the price is $24.556and you arrive open, you are better off if youclose the plant. (Of course, in this example, the value difference to keeping the plant mistakenlyopen is small. So, you cannot badly go wrong here.) In sum, your best choices are
If P = $24.556, If you arrive open, close the plant V = M$6.874
If P = $24.556, If you arrive closed, keep the plant closed V = M$7.074
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Figure III.5. Tree With Quarters Q2 to Q4 Worked Out
V means “value,” which includes both current revenue and discounted expected future revenues, assuming optimalfuture behavior, as well as opening and closing costs. One period revenue, if operating, is R(P ) = 400MW · (P −$25/MWh) · 2,191.5h. Present values (V ) include subsequent events, and are quoted in millions of dollars (M$). Ifyou arrive with an open plant, “Keep Opn” means your best action is to keep the plant open, “ Reclose” means yourbest action is to shut down the plant. If you arrive with a closed plant, “Keep Cld” means your best action is to keepthe plant closed, “ Reopen” means your best action is to open up the plant.
Section 3·3. The Simpler Method: Binomial Trees and Path Dependence.
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After much calculations, you have therefore learned that your action here is not a history-dependent decisions: the plant should not operate, no matter whether you arrive with an openor a closed plant (and no matter whether the price just increased or decreased).
Today
You are almost done. What should you do at the outset? If your plant is open, you keep it open The final valuecomputation! you aredone!!!!
and earn
V = +M$3.410 +1/2 · M$36.909+ 1/2 · M$7.074
1+ 2.5%= M$24.512 .
value today + disc exp value next quarter .(III.18)
If your plant needed to be opened up first, you would pay another $500,000. Figure ?? is yourfinal tree.
You could do further analysis. For example, you could determine what the expected cost of Additional Analysis:Expected ChangingCosts.
opening and closing is. Looking at the tree, there are 24 = 16 possible price paths. The followingprice paths require changing the state of the plant:
Price Path Action PV of Changing
UDDD close in Q4 $181,190
DUUU close in Q1, open in Q2 $671,029
DUUD close in Q1, open in Q2 $671,029
DUDU close in Q1, open in Q2 $671,029
DUDD close in Q1, open in Q2, close in Q4 $852,219
DDUU close in Q1, open in Q4 $648,097
DDUD close in Q1 $195,122
DDDU close in Q1 $195,122
DDDD close in Q1 $195,122
Therefore, the expected plant state changing costs are $267,497. This is fairly small—becauseyou will likely have to execute only a small number of plant changes. Now, compare your threevalue estimates:
Only quarterly changes permitted, changing costs M$25.012
Only quarterly changes permitted, zero changing costs M$25.284
In this case, the changing costs were modest, and the value estimate of the real option withchanging costs was not far from the solution without any changing costs.
Reflections — Heuristics as Substitutes
In one respect, this particular problem is pretty straightforward: it is an algorithm. You always “Easy” but tedious inprinciple. “Tough” inpractice.
work backwards, and keep track of the value of the plant in its possible states. It is just anexercise, perhaps tedious and computing-intensive, but something that you can solve with apen, a calculator, and a piece of paper—or by writing a computer program. In another respect,problems of this kind can very quickly become very complex. For example, how credible isthe price process? How can you handle a situation in which you have more than two choices,opening and closing? What do you do if there are 100 choices? (The answer is that at eachnode, you have to determine not just the best action given one of two states, but the best actiongiven one of 100 states. This is feasible if you write a computer program, but not if you haveto do it by hand.) And, what do you do if there is more than just the uncertainty about theelectricity price, but also uncertainty about other important inputs and outputs?
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Figure III.6. Tree With All Future Quarters Worked Out
V means “value,” which includes both current revenue and discounted expected future revenues, assuming optimalfuture behavior, as well as opening and closing costs. One period revenue, if operating, is R(P ) = 400MW · (P −$25/MWh) · 2,191.5h. Present values (V ) include subsequent events, and are quoted in millions of dollars (M$). Ifyou arrive with an open plant, “Keep Opn” means your best action is to keep the plant open, “ Reclose” means yourbest action is to shut down the plant. If you arrive with a closed plant, “Keep Cld” means your best action is to keepthe plant closed, “ Reopen” means your best action is to open up the plant.
Section 3·3. The Simpler Method: Binomial Trees and Path Dependence.
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Figure III.7. The Full Tree Worked Out — The Plant Value
V means “value,” which includes both current revenue and discounted expected future revenues, assuming optimalfuture behavior, as well as opening and closing costs. One period revenue, if operating, is R(P ) = 400MW · (P −$25/MWh) · 2,191.5h. Present values (V ) include subsequent events, and are quoted in millions of dollars (M$). Ifyou arrive with an open plant, “Keep Opn” means your best action is to keep the plant open, “ Reclose” means yourbest action is to shut down the plant. If you arrive with a closed plant, “Keep Cld” means your best action is to keepthe plant closed, “ Reopen” means your best action is to open up the plant.
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In some situations, you have no choice but to resort to simulations and heuristics—that is,Instead of workingsystematically backwards
to determine theoptimal choice, youcould just try out a
heuristic strategy.
imperfect operating rules. In our example, a heuristic may have stated “close an open plantif the price drops below $24.75,” and “reopen a closed plant if the price goes above $25.25.”(Your heuristics might also depend on whether it is early or late in the plant lease’s life). Almostsurely, the heuristic will occasionally recommend mistaken actions—it will sometimes tell youto operate when the perfect exact decision rule would tell you not to, and vice-versa. But theadvantage of most good heuristics is that they turn a path-dependent tree into a plain path-independent tree, where you no longer have to determine optimal decisions working throughthe entire tree. Instead, you can just work out the value at each node by itself. If you havewritten a program to just compute the expected value of the tree based on individual nodes,you can determine good policies through search—you can try out different heuristics (criticalshutdown and reopening prices) to see how the factory value changes. Of course, the betteryour heuristics are, the higher will be the value you attach to the factory.
Solve Now!
Q III.4 In Figure ??, work out the values in Q2.
Q III.5 Confirm the new values in Figure ?? that were not computed in the text.
Q III.6 Rework the tree if the closing cost is M$1 and the opening cost is M$2.
Q III.7 Rework the tree if the closing cost is M$4 and the opening cost is M$8. (Do this only if youhad trouble with the previous exercise.)
Q III.8 Rework the tree if the closing cost is M$16 and the opening cost is M$32. Do this only ifyou had trouble with the previous exercise.)
Q III.9 Compare the two benchmarks, the solution in the text, and the solutions in the threepreceding exercises. What would you expect?
3·3.D. Nerd Section: Valuation by Replication
Important: This section is not necessary to an understanding of how to valuereal options. If you want to understand this section, you must first thoroughlyunderstand how to value options and contingent claims in binomial trees. Thiswas explained in the Options and Derivatives chapter.
The techniques in this section can be helpful in some very specific situations—whenthe underlying uncertainty is already traded in financial markets. In such cases,it often provides more accurate valuations. Most corporate projects do not qualify,because few underlying uncertainties/commodities are traded.
Valuation by replication sees a project as providing payoffs that are contingent upon an un-Replication methods defacto obtain useful
project informationfrom market prices.
derlying commodity, if the forward prices (values) are known today. For example, as you shallwork out below, it might see your plant as a fancy set of derivatives on electricity forward con-tracts. If you know the electricity forward prices, you can value your plant using option pricingtechniques relative to the known electricity forward contract prices. The sole advantage of thistechnique is that you do not need to estimate your project’s own discount rate and the prob-ability of an up versus a down move. Instead, you get these handed to you by the techniqueitself. The market prices for electricity forwards implicitly determine the discount rate andup/down probabilities.
Section 3·3. The Simpler Method: Binomial Trees and Path Dependence.
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To make this clear, assume that when you check the 3-month Treasury bond, you find that the Assume a risk-free rateof 0% per quarter forcontrast to our owndiscount rate of 2.5%per quarter.
risk-free interest rate is 0%. This is lower than the 2.5% that your project commanded in theprevious sections—which you now officially forget, so that this section has a sense of purpose.Similarly, if you wish, you can now assume that you erroneously believe that the probability ofan up-move is 10%, rather than the correct 50%, which we have also officially forgotten. Thiswon’t matter, because this method does not use the up/down probabilities at all.
Assume that you are standing at the Q3 node where the current electricity price is $49.922/MWh. You need a forwardelectricity price, not thecurrent price.
Further, assume that in Q3 you can buy a forward contract on electricity that prices each unitof electricity in Q4 at $49.9218/MWh. (In other words, you can contract upfront to receive eachunit of Q4 electricity at this fixed price, rather than take the chance that the price may go upor down.) Note that the forward price need not be the same as the prevailing electricity pricein Q3!
Digging Deeper: Think about it: Oranges in spring are not the same commodity as oranges in summer. Infact, in spring, the immediate orange delivery price is usually much higher than the orange forward price forsummer delivery.
To value by replication, it is important that the “good next period” is storable. Forward contracts are piecesof paper, so they are easy to store. In Q3, the Q4 electricity forward contract is the same good as electricityin Q4. However, electricity in Q3 is not the same commodity as electricity in Q4—because electricity storage isprohibitively expensive. Of course, if the good itself is easily storable (e.g., like gold), it might be as almost goodas the storable forward contract. This is because if the commodity were freely storable, arbitrage determines theforward price from today’s commodity price.
If there is no forward contract available, and if the good is not storable or very expensive to store, valuation byreplication is not workable.
Do you need to know your project discount rate of 2.5% to determine the value of your plant Replicate payoffs bymaking sureforwards+bonds havethe same value in theup-state and down-stateas provided by the plant.
at the uppermost node (P = $49.922) in Q3? No, you do not! If you have δ forward contracts,each will be worth $59.906in the up-state, and $42.434in the down state. If you buy b treasurybonds, each will be worth b · (1+ rF) next quarter. Thus, if you purchase δ forward contractsand buy b risk-free bonds such that
δ · $59.906 + b · (1+ rF) = +M$30.599
δ · $42.434 + b · (1+ rF) = +M$15.282 ,(III.20)
then your forward+bond holdings will exactly replicate the payoffs of your plant. The solu-tion of these two equations are δ = 0.876 million forward contracts, worth M$$43.589, andb = −M$21.381. Confirm by substituting back: you will have +M$30.599 if the value of the0.876 million forward contracts were to go up from $49.9218/contract to $59.906/contract;and you will have +M$15.282 if the value of the forward contracts were to go down from$50.540/contract to $42.434/contract. Therefore, the value of your plant must be
Add to this the current Q3 revenues of M$21.847, and you determine that V = M$44.228—thesame value you obtained earlier, but without having to provide either the 2.5% discount rate orthe actual probability of up/down moves. You instead needed to know the electricity forwardprice in Q3 for Q4. Knowing one piece of information—either the forward price vs. the correctprobabilities and discount rates—is a perfect substitute for knowing the other.
This method is perfectly general. Your plant is just a set of options on electricity forwards. In Replication is atechnique for valuing aparent node.
fact, replication is merely a method of computing the value of a tree node from its two childrentree nodes in a different way from taking the probability-weighted discounted value. It haslittle to do with whether the project is currently closed or open, or whether the plant will beclosed or open (depending on state) in the next period. It also has nothing to do per sé withdetermining the decision of whether to keep the plant open or not (which depend on thesevalues but not on how these values are obtained).
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For example, if you are standing in Q3 and the current electricity price is $25.048, what is theAnother valuecomputation example. value of a closed plant? Substitute into
δ · $30.057 + b · (1+ rF) = M$3.933
δ · $21.290 + b · (1+ rF) = M$0.00(III.22)
to obtain δ = 0.449 and b = −M$9.551. If the forward price of electricity is $25.533, then thevalue of the plant is
Again, knowing the electricity forward price is a substitute for knowing the discount rate.
Important: Knowing the value of the forward contract on the underlying com-modity upon which the real option depends is a perfect substitute for knowing theproject’s own discount rate and probability of up/down moves in the tree.
There are many publicly traded futures and forward contracts—such as oil, pig bellies, oranges,Unfortunately, most ofthe time, you do not
know the value of theunderlying forward
contract.
copper, etc.—whose publicly known prices can be used to help determine the value of projectsthat primarily depend upon these commodities. Still, this is the exception rather than therule. Most corporate projects have revenues that are contingent on quantities for which you donot know the value of the underlying forward—in which case you shall have to return to youroriginal method of estimating up/down probabilities and a reasonable project discount rate.
Digging Deeper: The project discount rate depends on how the project cash flows covary with the overallmarket. Thus, your tree should write down at each point what the overall market value will be. Of course, thisreturns us to the world of the CAPM, with all its difficulties of estimating a good beta (which you now need to dofrom the tree), and more so on estimating a good market premium. Fortunately, you can often accept modestmistakes in discount rate estimates.
Finally, if the real option itself has no path-dependence—and very few real projects have none—Even Black-Scholes cansometimes be used! then you are back to a world in which the solution to the binomial tree can be much simpler.
Under some additional assumptions, you can even do a value by replication approach using thefamous Black-Scholes formula from the Options and Derivatives chapter.
Side Note: When managers claim that they have tried real-options methods, they often mean that they havetried to use the B-S formula—and often their approach is one in which the hammer is looking for the nail. Thatis, these particular managers are often recent egghead graduates who have a technique and are looking for aproblem to use it on, whether it is a good one or not. This is rarely a good idea: your choice of technique shouldbe driven by your problem, not the other way around.
Digging Deeper: Value estimates obtained from the B-S formula or any other option pricing technique (such asyour binomial tree) are usually very sensitive to your underlying volatility estimate. The volatility input—whichdetermines the up/down ratios in the binomial tree—essentially comes from your estimate of the underlying(electricity price) process. Estimating volatility correctly is hard, and if you make mistakes, your value estimatecan be far off.
But what about the non-replication approach, where you need the discount rate for your project and the probabilityof up/down moves? Is it more robust? Your simplification of a constant discount rate of 2.5% at each nodewas almost surely incorrect—at some nodes you face less risk than at other nodes. For example, if the price is$17.742/MWh in Q4, you know you will not operate and get the same constant cash flow, no matter what, so therisk-free interest rate would have been enough. Similarly, if the price is $25.048/MWh, the discount rate will beless than if the price is $49.922/MWh. Estimating the proper discount rate and proper probabilities of up anddown moves is hard, and if you make mistakes, your value estimate can be far off.
You are caught between a rock and a hard place. Valuation is difficult!
Section 3·4. The Sophisticated Method: Time-Series Processes and Monte-Carlo Simulation.
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3·4. The Sophisticated Method: Time-Series Processes andMonte-Carlo Simulation
You now know how to solve real option problems with pencil and paper if you can write down You can allownon-binomial processes.a good electricity process in a binomial tree. Can you do better than the binomial process if
you use heavy statistical artillery to model historical electricity prices, and then use the modelto predict the future electricity price? Probably. This is the first goal of this section—we wantto develop more general techniques to estimate future electricity prices, upon which your realoption valuation ultimately depends. To fit into this one chapter, we now assume that electricityprice changes are “only” weekly.
We will rely on three computing intensive tools, which must be applied in sequence: The 3 main tools.
1. Time-Series Data Modeling (to understand the historical electricity price process).
2. Monte-Carlo Simulation (to model the future electricity price process).
3. Optimization (to choose the best policy), more specifically Monte Carlo based optimiza-tion.
Off-the-shelf spreadsheets are not especially well suited to any of these tasks, but they areflexible enough to be contorted to work each one. (The example spreadsheet used for thisproblem is on the book’s website at http://welch.econ.brown.edu/book.) Morevoer, there are anumber of vendors selling spreadsheet add-ons to make these three processes simpler. Evenbetter tools leave the realm of spreadsheets and/or require programming—the best would bea statistical program for time-series data modeling, a program for Monte Carlo simulation, anda programming library for optimization within the Monte Carlo simulations.
3·4.A. Statistical Data Modelling: Understanding The Historical Electricity Process
Statistical modeling is not the domain of finance, but the domain of statistics. This domain is We can only do a “tour”of statistics.so large that there are literally thousands of books and professors, showing how it can be done.
This chapter can only give you a taste of what can be done. Of course, if you are truly betterthan your competition in modeling time series, then you can predict the future better—andthus probably get very wealthy very quickly. An ability to forecast better would be a huge skill.
Figure III.8. Average Weekly Wholesale Price of Electricity Per MWh at the California PowerExchange in 1999.
Your first task is to settle on a suitable model for electricity prices. Figure III.8 plots theHere are 4 modelingoptions. historical time-series from Table III.1. How can you model this process? You have many choices,
among them the following four:
1. You can model electricity prices as being drawn from a stationary distribution:Price is constant plusnoise.
Pt = a + noiset−1,t . (III.24)
This process is equivalent to drawing a straight horizontal line in the graph. Table III.1shows that your best estimate for awould be its historical mean of $28.09, and you wouldbelieve that the noise has a standard deviation of $10.11.
This process does not fit the historical data: For example, prices in February were allclustered around $18-$19. Prices in November and December were mostly about $30. Ifformula III.24 were accurate, it would be highly unlikely that your historical prices wouldhave been as persistent as they were. Thus, you can conclude that this is probably not agood process to describe history.
2. You can model prices as being driven by a trend:Price is constant andtrend plus noise.
Pt = a + b ·Week Numbert + noiset−1,t . (III.25)
This process is equivalent to drawing a slightly diagonal line from about $18 in the firstweek to about $40 in the last week.
Although better, this process does not fit your data too well, either. It is true that theending price was higher than the starting price, but the trend is not that of a constantincrease. Specifically, after October, the price fell and staid low. (But even in earlierweeks, such as week 30, your best forecast would likely not always be a point on thestraight diagonal line. Instead, prices seem to be “sticky.”)
3. You can model prices as being driven by some third variable that you know today. ForPrice is driven bysomething else you
already know.example,
Pt = a + b · Known Stock Pricet−52 + noiset−1,t . (III.26)
Alas, you have neither reason to believe that the stock price one year ago is indicative ofthe future electricity prices, nor is it likely that you know any good variables today withwhich to predict future electricity prices.
(However, there are situations in which other historical variables do predict well. Forexample, an airplane takes one year to build. Purchases of airplane parts by manufacturerscan be good predictors of finished airplane production one year later.)
4. You can model prices as being related to past prices. Such processes are also often calledAR processes are oftenthe best modeling choice
for time-series.univariate time-series processes, because there is only the one variable itself in the model.A common univariate time-series specification is the auto-regressive process (AR pro-cess), in which the current value depends on past values:
You have already seen at least one such process: a simple random walk is Pt = 0+1·Pt−1, aspecial form of an AR(1) process. Superficially, this process seems to fit electricity prices:weeks in which the prices were high (low) are more likely to be followed by weeks in whichthe prices were still high (still low). Figure III.9 makes this even easier to see: it plots thecurrent electricity price against last week’s electricity price. Indeed, it appears as if theweekly price does have an influence on the next weekly price.
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Let’s settle on a simple AR(1) autoregressive process, called AR(1) because we posit that only We now specify theregression to estimatethe AR process.
the most recent price (1 lag) has any influence:
Pt = a + b · Pt−1 + noiset−1,t . (III.28)
To determine the best coefficients a and b, you must estimate a linear regression. (In Excel,this requires loading the Analysis Toolpak in the Tools/Data-Analysis menu. In OpenOffice, youcan use the LINEEST facility.) In regression terminology, Pt is the dependent (Y ) variable, andPt−1 is the independent (X) variable. So, create a column next to the actual electricity pricethat is the past electricity price. (The column is shifted down one, with the end points suitablytruncated.)
your Y your X
↓ ↓A B C
1 Week Cur Prc Lag Prc
2 Jan1-Jan2 12.89 #N/A
3 Jan3-Jan9 23.14 12.89
4 Jan10-Jan16 24.13 23.14
5 Jan17-Jan23 20.78 24.13
6 ... ... ...
You can now instruct your spreadsheet to estimate a regression with Y being the B-column and Excel can estimate goodAR(1) parameters.X being the C-column. The Excel regression output is in Table III.2.
Figure III.9 plots the price of electricity as a function of the past price. The fitted AR(1) line(y = 8.92 + 0.69 · x) is also plotted in the figure. The figure confirms that there is a strongrelationship between the lagged electricity price and the current electricity price.
Figure III.9. Electricity Price Vs. Previous Electricity Price.
20 30 40 50 60
20
30
40
50
60
elec.price.lag
elec
.pri
ce
You second step is to estimate the variability of your forecast error. The more variable the noise,Collect what you needfor the next step:Second, the noise
variability.
the more valuable your real option. (The mean of noise is always 0.) The regression outputinforms you that the noise has a standard deviation of 7.1058... . What do your residualslook like? Figure III.10 plots them. You only have 51 data points, so the histogram is notperfect. Still, with some imagination, you can see that your electricity residual seems to followa somewhat bell-shaped distribution. Thus, you may take the liberty of assuming that yournoise is distributed as if it were drawn from a normal distribution with a mean of 0 and astandard deviation of 7.1.
Digging Deeper: There is also other interesting price information in the regression output. The R2 tells usthat you can explain about 50% of the variation in the electricity price. The t-statistic on the X-variable is above7, indicating very high statistical significance. (PS: We are ignoring some special issues with time-series process,called stationarity, which will prevent the coefficient estimate on Pt−1 to approach 1 even if the underlying processhas a coefficient of 1.)
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Figure III.10. Histogram of AR(1) Residuals (Noise).
Value
Freq
uen
cy
−10 0 10 20
05
10
15
20
Digging Deeper: Your model is probably not the best process. First, you did not explore further laggedprice terms. However, exploration shows that this happens not to matter: further lags have no extra significance.Second, your relation in Figure III.9 exhibits heteroskedasticity: that is, when the electricity price is high, theresidual also tends to be high. You could improve both your coefficient and risk estimation if you took this intoaccount. For simplicity, we won’t. Third, you are ignoring seasonalities. (Of course, here you do not have thedata from earlier years to model it, anyway.) Fourth, even in your process, you are actually underestimatingyour uncertainty: you also have parameter uncertainty (about a and b), and farther out prices tend to have moreparameter uncertainty. Fifth, your process allows the electricity price to become negative. This is an avoidableflaw: A better alternative may be to estimate the price process in logarithms instead of levels. Here is how youcould do this: Estimate
log(PtPt−1
)= a + b · log
(Pt−1
Pt−2
)+ c · log
(Pt−2
Pt−3
)+ ... + noiset−1,t . (III.30)
A good process is estimated to be
log(PtPt−1
)= 0.0087 + (−0.279) · log
(Pt−1
Pt−2
)+ noiset−1,t , (III.31)
where the noise has a residual standard error of 0.2179. To use this model to predict, start with the two finalprices in the series, $28.31 and $28.89. The log of their ratio is 0.02028. Thus, your prediction would be
Should you put all faith in this “deus ex machine” of modeling? No. In many situations, there is Use “soft” informationto improve quantitativeforecast.
inadequate historical data to build a quantitative model of the future. In other situations, real-world experience and qualitative judgment are outright better tools than quantitative modeling.But most often, the best practice combines practical experience with quantitative modeling. Itis not difficult to informally incorporate “soft” qualitative information into your “hard” quan-titative price process. For example, you may have additional information that is not containedin historical prices. You may know that Congress has just passed an energy subsidy, which willlower the electricity price by $1/MWh. You could use this knowledge to reduce the constantfrom 8.92 to 7.92. Or, you may know that next year, a new uptrend in electricity prices may
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occur that will increase the time-series coefficient by 0.05. Or you may know that next year’selectricity price uncertainty will be higher than this year’s, so you may choose to increase thestandard deviation from 7.1 to 10.0.
Solve Now!
Q III.10 Is it typically better when an estimated process has residuals that have lower variance?
Q III.11 Estimate the constant model and the trend model from the text. Do they fit better? Dotheir residuals have more or less variance than the AR(1) process?
Q III.12 Estimate an AR(2) model. Does it fit better? Do its residuals have more or less variancethan the AR(1) process?
3·4.B. Monte Carlo Simulation: Modeling The Electricity Price Future
You now have an estimate of the time-series process driving electricity prices. Your next stepYour starting point isthe electricity
time-series process.is to use your process to simulate a model of the future—a testbed upon which you can testout operating strategies.
Both Excel and OpenOffice have built-in random number generators. The functions are calledHow to generate randomnumbers that are
(Gaussian) normal.RAND(), and return a uniformly distributed random number. To produce a random numberfrom a bell-shaped normal distribution, you need to use the NORMSINV() function, which trans-lates the uniform random number into a normally distributed random number with a mean of0 and a standard deviation of 1. You want the noise to have a standard deviation of 7.1, so youmust multiply it by 7.1. So the formula to obtain a single simulated residual is
= 7.1 ·NORMSINV(RAND()) . (III.34)
If you were to repeat this a million times, you would find that the distribution of these valuesindeed looks like a bell-shaped curve with a mean of 0 and a standard deviation of 7.1.
You can now generate possible future price scenarios, each being one column. Each scenarioHow to generate a singleprice scenario (all 52
weeks).starts with the last known price in 1999 of $28.89in the first row. The next price, in row 3below, is the formula
Some sample future price scenarios so generated might be:
A B H N T
1 Week Scenario 1 Scenario 2 Scenario 3 ...
2 12-1999 28.89 28.89 28.89 ...
3 1 23.54A 33.92B 35.72B ...
4 2 36.28B 21.83B 31.84B ...
5 3 41.01B 38.89B 26.53B ...
A The formula in this cell is: = 8.92+ 0.69 · B2+ 7.91 ·NORMSINV(RAND()).B The formula in this cell was copied as a formula from A. The spreadsheet automatically changes the reference.
Upon a recalculation (the F9 key), the spreadsheet will draw new random values (due to theRAND() function use), and different price scenarios will appear.
Section 3·4. The Sophisticated Method: Time-Series Processes and Monte-Carlo Simulation.
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Solve Now!
Q III.13 Replicate the spreadsheet in Table ??. Stare at what happens when you recalculaterepeatedly ( F9 ).
3·4.C. A Detour: Comparing Electricity Price Processes
You now have two electricity price processes for the future: the simple binomial one, posited Comparing the binomialprocess with thetime-series estimatedprocess.
in the previous section, and the time-series estimated process, posited in this section. How dothey differ?
Figure III.11 plots the distribution of the average electricity price just in the final fifth quarter.The line with the smooth spikes is your binomial process; the squiggly line is the simulation ofthe weekly time series process (Formula III.29 on Page 82). Note that it can be rewritten as
which states that the price is a weighted average of the last price and $28.77. Thus, the long-runtendency of the electricity price is to return to $28.77, but only about 31% of any deviation from$28.77 is typically corrected in the following week. Such a process is called mean-reverting,and it is intuitively the reason why the electricity price tends not to wander too far away from$28.77.
Inspection of Figure III.11 shows that, although the two processes have similar means, medians, The mean reversioncauses lower variance!and most-likely price in the fourth quarter, they nevertheless differ drastically: the binomial
process is a lot more variable than the time-series process. Thus, your new electricity priceprocess—from the time-series estimation in this section—will cause your estimate of the valueof the plant to be lower than it was under the binomial process of the previous section. Thereal option to shut down and reopen once per quarter will be estimated to be less valuable.
Solve Now!
Q III.14 How would Figure III.11 look if the process were
Q III.15 How would Figure III.11 look if the time-series process had noise with a standard devia-tion of $1 instead of $7.1?
Q III.16 Warning: Tedious. Warning: Solve only if you know statistics. Set the electricity pricechange in a down move to 0.89 (a−11% change), and the up price move at 1.11 (+11%). Computethe electricity price mean and standard deviation of this process in the final quarter. Repeat thebinomial tree valuation.
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Figure III.11. Comparing Processes’ Predicted Electricity Prices in the 5th Quarter
20 30 40 50 60
0.0
00.0
50.1
00.1
5
Average Electricity Price, 5th Quarter
Den
sity
The smooth spikes represent the binomial distribution used in the previous section. Its mean is $31.89, its standarddeviation is $11.13.The squiggly distribution is from 50,000 simulations of the time-series estimated process Pt ≈ 8.92+ 0.69 · Pt−1 +noiset−1,t , where the noise has a standard deviation of $7.1. In the final quarter, the average electricity price is$28.77 with a standard deviation of $6.31..
3·4.D. Exploring the NPV of One Particular Heuristic Strategy
As a manager, what strategies can you choose? Your managerial strategy is a rule that decidesHow to model yourbehavior: a managerial
strategy is a pair ofcritical prices.
whether to operate or not to operate your turbines. You can condition your choice on thecurrent state of the plant and on the current electricity price. For example, you may follow the“heuristic” strategy
If the plant... Operate If Close If
was running (t − 1) Pt ≥ $24 Pt < $24
was shut (t − 1) Pt ≥ $26 Pt < $26
Table III.3 shows how you obtain the plant NPV given this strategy and given one particularprice scenario. Note how we keep the critical strategy values in D1 and D2, so that you caneasily try other strategies later.
In the C column (next to the price column B), you keep the current state of your plant. ItKeep track of the plant’soperating state. contains the formula
If(Ot−1 = 1, then If(Pt > 24, then1, else0),
else If(Pt > 26, then1, else0)) .(III.39)
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In the D column, you keep track of incurred reopening costs ($500,000) and incurred reclosing Keep track of operatingstate change costs.costs ($200,000) with the formula
If(Ot−1 = 1, then If(Ot = 0, then500,000, else0),
else If(Ot = 1, then200,000, else0)),(III.40)
In the E column, you compute the revenue (0 if you are not operating), Keep track of electricitysales. Then add it all upand discount itappropriately.If(Ot = 1, then7 · 24 · 400 · (Pt − $25), else0) . (III.41)
In the F column, you keep the appropriate discount factor. Assume that the beta of the plantwith the stock market is such that the appropriate quarterly discount rate is 2.5%.
Dt =Dt−1
(1+ 2.5%)1/13. (III.42)
The G column contains the (suitably discounted) cash flows to obtain the value, given yourassumed simulated price at this point, and given your particular strategy heuristic. The sumof all numbers in the G column is the net present value of the cash flows for five quarters (65weeks), given your simulated price process.
Each time F9 is pressed, the spreadsheet recalculates: it draws a new set of random numbers, Each random pricescenario gives adifferent NPV outcomefor a specific strategy.
which ultimately results in a new NPV in cell G60. The average of many (thousands) of suchprice scenarios is the expected NPV of your plant given the one heuristic strategy (opening andclosing critical prices) that you are exploring. How many different price path scenarios shouldyou entertain? Of course, the more the merrier. But, are 100 enough? Are 1,000 enough? Thisdepends on the situation and the computing power available. In our case, about 10,000 drawsgenerate reliable values. But even only 100 draws (possible futures) are better than assumingthat there is only one possible future!
Alas, writing down the many thousands of possible NPVs to find the correct NPV of the pro- Writing a spreadsheetmacro?ject is painful. You need some automation: you want to recalculate, and then store away the
single outcome price-scenario G72, so that you end up with thousands of single outcome price-scenario G72 calculations, over which you can compute the grand average. You have threechoices:
• You can learn the OpenOffice macro language, and/or rely on the OpenOffice macro inAppendix Section a below. (This is my own choice.)
• You can learn the Excel macro language, and/or rely on the Excel macro in AppendixSection b below.
• You can spend $1,500 to purchase an add-on for Excel called Crystal Ball, which doesthese simulations in a much more convenient manner.
Digging Deeper: This technique also allows keeping track of other interesting statistics. For example, youmay want to know how frequently you shut down or reopen your plant. Or you may be interested in how oftenyou experience cash crunches. For example, you can build a spreadsheet that starts with a cash balance of say,$1 million, and each week adds current revenues. If you go below a critical value, the company would be declaredcash-crunched, and you could determine how often you would end up in such a bankrupt state.
Solve Now!
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Table III.3. Single Price Scenario Valuation under Given Heuristic Strategy
A Known Starting Value, $28.89B Formula: = 8.92+ 0.69 · B5+ 7.91 ·NORMSINV(RAND()).
B formula is your modelled price process.CF Copied Formula From B (The spreadsheet updates the references).D Known Starting Value, Plant Operated.E Formula: = if(C5 = 1, if(b6 < $D$1,0,1), if(b6 > $D$2,1,0)).
E formula says “if you operate, and price is too low, stop;if you do not operate and price is too high, start.
F Copied Formula from E (The spreadsheet updates the non-dollared references).G Known Starting Value: 0. (No Cost)H Formula: = if(C6 = 0 and C5 = 1,−$G$1, if(C6 = 1 and C5 = 0,−$G$2,0))
H formula says if you just switched plant state, you have to pay.I Copied Formula from H (Spreadsheet updates non-dollared references).J Formula: = if(C5 = 1,7∗ 24∗ 400∗ (B5− 25),0).
this is operating revenues IF you operate.K Copied Formula from J (Spreadsheet updates references).L Known Discount Factor Value Today: 1.M Formula: = F5/(1+ 0.001901237).
each week, the discount factor declines by (1+ 2.5%)1/13.N Copied Formula from M (Spreadsheet updates references).O Formula: = F6∗ (D6+ E6).
PV is sum of revenues and opening/shutting costs.P Copied Formula from O (Spreadsheet updates references).Q Formula: = sum(G6 : G70).
Q sum is one simulation NPV, not the average NPV!
Sidenote: the only differences between Excel and OpenOffice are [1] that the latter uses semicolons as separators inthe “IF” statement, while the former uses commas; and [2] the logical “AND” is written as C6=0 and C5=1 in Exceland as AND(C6=0;C5=1) in OpenOffice.
Section 3·4. The Sophisticated Method: Time-Series Processes and Monte-Carlo Simulation.
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3·4.E. Optimizing For the Best Managerial Strategy by Trial-and-Error (Monte Carlo)
You now have all the necessary tools to find the best strategy. Each strategy is a pair of critical Try many differentstrategies to determinea good strategy.
prices (at which you reopen, and at which you reclose). Given a strategy, you know how toobtain a plant NPV, using many thousand simulated price scenarios. Thus, you can now try outdifferent strategies, and experiment until you find the strategy pair that yields the maximumNPV. It is relatively easy to try out broad strategies such as ($25,$25), ($24,$26), ($24,$30), etc.,to get a good idea of an approximate reasonable strategy. With 50,000 price scenarios, my ownaverage NPV results for some “symmetric” strategies—symmetric because the critical pricescenter around your cost of $25—are in Table III.4.
Table III.4. NPV and Frequency of Plant Changes under Various Heuristic Strategies
Symmetric Strategies
Reclose if Reopen If Strategy Frequency of
price is below price is above NPV Close/Reop
$25.00 $25.00 $22.5 24%
$24.00 $26.00 $23.1 21%
$23.00 $27.00 $23.6 18%
$22.00 $28.00 $23.8 16%
-→ $21.00 $29.00 $23.9 14%
$20.00 $30.00 $23.8 12%
$19.00 $31.00 $23.6 10%
$18.00 $32.00 $23.4 9%
$17.00 $33.00 $23.0 8%
$16.00 $34.00 $22.6 7%
$15.00 $35.00 $22.2 6%
$10.00 $40.00 $19.7 3%
$5.00 $45.00 $17.5 1%
$0.00 $50.00 $16.6 0%
If you try out additional price pairs not in Table III.4, you will find that the best symmetric strat- Do not look for toomuch accuracy.egy is about $21.08 and $28.92 in my simulations, producing $23.9 million in NPV. Fortunately,
the difference between a strategy of ($21.00,$29.00) and even ($22.00,$29.00) is typically rel-atively modest, which means that there is no need to optimize the strategy down to the lastcent—your price model uncertainty is considerably worse than your Monte Carlo accuracy.
Incidentally, the last column in Table III.4 is an unnecessary “fun statistic” that you can keep Other fun statistics.
track of: it describes the frequency with which you see a change in operating procedure (aclosed plant reopening or an open plant reclosing). If you ignore reopening/reclosing costs—that is, your strategy asks for shutting down and reopening at P = $25—you are changing plantstate in about one in four weeks. If you are smarter, at your optimal strategy, you would incurthe reopening/reclosing costs only in about 14% of all weeks.
Digging Deeper: There are some numerical optimization pitfalls. This is due to the random nature of theprice path scenarios—it is better to compare multiple strategies on the same price paths than each strategy on itsown set of price paths.
We are pretty much done. Let’s just recap the performance of difference strategies and scenar- Different strategyscenarios in overview.ios. If you set the costs of reopening and reclosing in your program to zero, you end up with a
non-path-dependent plant value of $27.8 million. Similarly, if you set the cost of reclosing theplant to a huge number, you end up with a non-path-dependent plant value of $15.9 million.You have already computed the optimal strategy, given changing costs, above as shutting downif the price falls below $21 and reopening if it goes above $29: the NPV is $23.7 million.
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Side Note: For “sick” fun, I wrote a program that estimates the value if you allow only quarterly plant changes,so that in comparing with the value estimates from the previous section, you can see what is due to our abilityto change more often, and what is due to our different price processes. This value is $16.2 million, because themean reversion makes it very rarely optimal to close down for a full quarter even at a low price. This is muchbelow the $25.197 million from the previous section, so you can conclude that our revised price process playsa very important role in our value estimate.
Always Operate Plant $15.9 million
Quarterly Changes Permitted $16.2 million
(Changing is rarely optimal, due to the mean-reversion)
Weekly changes permitted, with changing costs $23.7 million
($21.00 and $29.00 critical values)
Weekly changes permitted, no changing costs $27.8 million
Solve Now!
Q III.17 Replicate the table with the ultimate payoffs. Are your results exactly the same?
Q III.18 Why is the NPV of $23.7 million less than the $28.6 million that you computed in aback-of-the-envelope fashion in Section 3·2.B?
Q III.19 What is the average NPV for the ($15,$25) strategy; i.e., if you shut down an operatingplant when the electricity price drops below $15, and reopen a shut plant when the price is above$25? Is it better or worse than the ($25,$25) strategy?
Q III.20 What is the average NPV for a ($13,$27) strategy; i.e., if you shut down an operatingplant when the electricity price drops below $13, and reopen a shut plant when the price is above$27? Build a spreadsheet to explore this strategy.
3·4.F. Discussion
The three techniques—statistical modeling, Monte Carlo simulation, and optimization—canThe techniques fromthis chapter are
powerful. Use them for:handle a wide range of problems. Here are some examples:
• You could replace the price process with a more sophisticated model that takes intoBetter processestimation. account such factors as seasonalities, the reality that prices cannot go negative, non-
normal error distributions, etc.
• You could value different kinds of real options. For example, the factory may not beOther types of realoptions. reopenable for 1 month once shut down. Or, you might be able to produce more output
by adding another $300,000 in maintenance per week. Each of these constraints/optionswould require clever additional “if-then” expressions in the spreadsheet to model them.But all modeling principles would remain the same.
• You could explore non-symmetric strategies (in which the price to reopen the plant mightOther kinds of strategies.
have been $30 and the price to reclose it might have been $24). The computational opti-mization complexity would increase, but the methods would remain the same.
Section 3·4. The Sophisticated Method: Time-Series Processes and Monte-Carlo Simulation.
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• You could not only model the electricity price as a univariate time series process, but as Multivariate processes(e.g., to estimate beta).a process that has a correlation with the stock market. (Perhaps the stock market goes
down when electricity prices rise.) In this case, you would have an additional column withsimulated stock market returns, and you could determine what the correlations of yourstrategized cash flows with the stock market would be. This would eventually give you amarket-beta, which would help you improve on your cost of capital estimate (here, 2.5%per quarter).
• You could model your cash position. For example, you may start with $1,000,000 in cash, Keeping track of cashand avoiding cashcrunches.
and the operating cash flows net of transaction costs cumulate into your cash position.If your cumulative cash balance ever falls below $0, you could introduce additional costs(either to borrow more cash, or to model your bankruptcy costs). You could then usethis to determine whether you really need $1,000,000 in cash, or whether you can settlefor $500,000. (Monte Carlo methods are probably the right way to think about managingworking capital!)
• You could also easily obtain estimates of the uncertainty. For example, how likely is it Estimate project risk(losing your job!).that you will end up with an NPV that is below $20 million? More than $30 million? This
is easy to compute: instead of computing the mean NPV over your 50,000 price scenarios,just count the fraction of outcomes that satisfy your criterion.
The combination of these three techniques is tremendously powerful, but using them requires OK, so the techniquesare not easy. But theyare flexible and oftenextremely useful.
a good amount of sophistication and experience. However, do not be deceived: there aremany real-world problems that may be in principle solvable with these techniques, but thatin practice may require more time to solve than the universe has seconds. (This typicallyhappens with multi-dimensional sources of uncertainty and multiple strategy choices.) So, inthe end, these “quantitative” techniques often help with critical choices, but only after economic“qualitative” knowledge has been used to suitably simplify the problem. And, never forget themost important test: have you used common sense in your model inputs, and does the modeloutput make sense to you? If not, dive deeper into your model. Never use a black box that youdo not understand and that does not make sense to you.
Solve Now!
Q III.21 Assume that you start with a cash position of $550,000. How likely is it that you will runout of cash in any one week, if you follow strategy ($25,$25) vs ($20,$30) vs. ($15,$35)? Assumeyou earn no interest.
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3·5. Summary
The chapter covered the following major points:
• Real options are embedded in real production activities. They are the value of your futureflexibility.
• Real options are more valuable when there is more uncertainty about the future.
• Combine both quantitative and qualitative information when assessing the value of realoptions and when simulating plausible futures. Some problems are more suited to theformer (e.g., uncertainty for which you can get data from a long history); other problemsare more suited to the latter (e.g., a mission to Mars).
• You can work many real option problems with a binomial tree. The two rules to rememberare “work backwards,” and “keep track of the current state of your plant.”
• Path-dependent trees are more complex, because the best action at any given node de-pends not only on the current node, but also on the state of the asset (which depends onthe path you took in the tree). Path dependence often arises if there are costs to changing.
• Valuation by Replication is not a requisite tool to valuing real options. However, in theoccasional case where forward contracts on the underlying risky basis are available, it canbecome a useful alternative to estimating one’s own discount factor and future probabil-ities of better/worse times. This can make value calculations in the binomial tree morereliable.
• If you abandon binomial trees, you can instead rely on more general modeling of theunderlying uncertainty. That is, you can use statistical forecasting to estimate both meanand uncertainty.
• You can model/simulate the future, given the statistical properties that worked in the past.You can always informally add qualitative knowledge to your quantitative estimates.
• Scenario analysis can help you understand the range of possible future outcomes. Eachscenario is the outcome in one particular future price scenario.
• Monte Carlo analysis can automate scenario analysis.
Section A. Advanced Appendix: Monte Carlo Repeated Drawing.
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Appendix
A. Advanced Appendix: Monte Carlo Repeated Drawing
The following are spreadsheet macros that instruct their spreadsheets to recalculate a formula,and copy the results of each scenario into a range of cells. The formula itself can be as complexas the user desires, as it can be copied from spreadsheet calculations anywhere. The numberof iterations in these particular macros is handled differently: it can be set in the macro inOpenOffice and in the spreadsheet itself in Excel. The functionality of the macros is basicallythe same.
a. An OpenOffice Macro
REM ***** BASIC for OpenOffice calc *****REMREM (C) Ivo Welch, 2004.REMREM The input NPV must be in cell $G$72. (you can change this.)REMREM Each individual simulated NPV is put into cell $I$73, and earlierREM simulations will be pushed downward (beginning with $I$73). Thus,REM the function can be executed as often as desired, and more simulationREM outputs will thereby be added. To restart with a new strategy orREM simulation, just erase the entire column I with all its numbers.REMREM Your spreadsheet can then compute the mean and standard deviationsREM of column I, as is done in the sample spreadsheet provided on theREM book’s website.REMREM You can also download this macro by downloading the OpenOfficeREM spreadsheet used to work our electricity plant example.
REM ----------------------------------------------------------------------REM run_500_simulations makes this faster.
sub run_500_simulationsfor counter=1 to 500
record_one_simulation_runnext counter
end sub
REM ----------------------------------------------------------------------sub record_one_simulation_run
REM ---- ooffice bookkeeping: define variablesdim document as objectdim dispatcher as objectREM ---- ooffice bookkeeping. get access to the documentdocument = ThisComponent.CurrentController.Framedispatcher = createUnoService("com.sun.star.frame.DispatchHelper")
REM ---- now our own macro begins. first, go to cell $G$72 and copy it.dim args1(0) as new com.sun.star.beans.PropertyValueargs1(0).Name = "ToPoint"args1(0).Value = "$G$72"dispatcher.executeDispatch(document, ".uno:GoToCell", "", 0, args1())dispatcher.executeDispatch(document, ".uno:Copy", "", 0, Array())
REM ---- now, go to $I$72 and copy it;dim args3(0) as new com.sun.star.beans.PropertyValueargs3(0).Name = "ToPoint"
REM ---- make it a dollar formatted figuredim args6(0) as new com.sun.star.beans.PropertyValueargs6(0).Name = "NumberFormatValue"args6(0).Value = 103dispatcher.executeDispatch(document, ".uno:NumberFormatValue", "", 0, args6())
REM ---- finally, paste it.dim args5(0) as new com.sun.star.beans.PropertyValueargs5(0).Name = "Flags"args5(0).Value = "V"dispatcher.executeDispatch(document, ".uno:InsertCell", "", 0, args5())
end sub
b. An Excel Macro
’ Ken Wieler was kind enough to create this Excel macro for me.Sub RunMonteCarlo()
Dim intStartRow As Integer, intEndRow As Integer
’Read in Start and End ValuesstrStartCell = Range(‘‘B1’’).ValuestrEndCell = Range(‘‘B2’’).Value
’Find out Column and make sure the start and end are in the same columnstrStartCol = UCase(Left(strStartCell, 1))strEndCol = UCase(Left(strEndCell, 1))If strStartCol <> strEndCol Then
Exit SubEnd If
’Blank the range of Cell’sRange(strStartCell, strEndCell).Value = (")
Do While intStartRow <= intEndRowstrWriteCell = strStartCol & intStartRowstrValue = Range(‘‘B3’’).ValueRange(strWriteCell).Value = strValueintStartRow = intStartRow + 1
LoopEnd Sub
In the Excel sample output below, you merely compute the NPV if the outcome from operatingnext year is uniformly distributed between $400 and $600, but a guarantee exists that offers$500 as the minimum. The spreadsheet calculates that at a discount rate of 5%, the NPV of thisproject is around $505, although the standard deviation is plus or minus $30.
Section A. Advanced Appendix: Monte Carlo Repeated Drawing.
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A B C D
1 Your Starting Cell is: D1 $476.19
2 Your Ending Cell is: D20 $490.88
3 To-Be-Copied Result $489.40 ’← =B5; think “transfer station” $555.32
4 $543.40
5 My Result (e.g., NPV) $489.40 ’← =max(b7,b8) $476.19
6 $476.19
7 If Operating $513.87 ’ ← a random variable, uses rand() $536.20
8 If Not Operating $500.00 $476.19
9 Discount Rate 5% $476.19
10 $527.91
11 $476.19
12 $476.19
13 $542.77
14 $507.82
15 $476.19
16 $540.24
17 $476.19
18 $549.21
19 $535.82
20 $476.19
21
22 Expected Value → $504.57
23 Standard Deviation → $31.29
Entries in C3–C7 describe entries to the left; entries in C22–C23 describe entries to the right.
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c. A C++ Program To Work Out the Binomial Tree
In case you are a computer programmer with good knowledge of C++, you may enjoy thefollowing recursive program that solves for the price of the plant in the binomial case.
#include <stdio.h>#include <assert.h>
//// Sep 2005: This is a recursive function that solves the binomial// tree example in the real options chapter. The answer is something// like 25.01168.//
/****************************************************************/const double presentvalue(const double price = 28.89, const int state =OPEN, const int quarter =0,
int main() {printf("The present value is %.5f\n", presentvalue());return 0;
}
Section A. Advanced Appendix: Monte Carlo Repeated Drawing.
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Solutions and Exercises
1. Yes. If the variance is sufficiently high, your real option may be more valuable.
2. At $28 production cost, at an average output price of $28.09, the first technology would have earned us 9cents per MWh, which would come to $315,360 in revenues. At $29 production cost, there would have been18 weeks where production was worthwhile ($32.26, $40.80, ..., $31.26), with an average price of $39.13. Thus,the revenues would have been 400 · 18 · 7 · 24 · $10.13 ≈ $12 million. Clearly, in the typical week, the firsttechnology would provide a better outcome, but it is not the higher PV technology!
3. Variable cost.
4. Do it! As an aid, if P = $29.468, the current net revenue is R(P) = +M$3.916. The expected discounteddownstream value if operating is either $18.700 or $2.107, for an expected net revenue of M$10.403, whichdiscounts to M$10.150. Therefore, the value is M$14.066.
5. The correct answers are already in the figures. For P = $29.468, if you arrive open, you have a value ofM$14.018, which is
value today + disc exp value next quarter .(III.43)
6.
Quarter PriceIncoming
StateOptimalAction R(P ) V
Q0 $28.89 Is Open Keep Open M$3.41 M$24.625
Q1 $24.556 Is Closed Keep Closed M$0 M$6.075
Q1 $24.556 Is Open Keep Open -M$0.389 M$6.174
Q1 $34.668 Is Closed Open Up M$6.475 M$35.316
Q1 $34.668 Is Open Keep Open M$8.475 M$37.316
Q2 $20.873 Is Closed Keep Closed M$0 M$0.579
Q2 $20.873 Is Open Close Down -M$1 -M$0.421
Q2 $29.468 Is Closed Open Up M$1.916 M$11.876
Q2 $29.468 Is Open Keep Open M$3.916 M$13.876
Q2 $41.602 Is Closed Open Up M$12.553 M$43.249
Q2 $41.602 Is Open Keep Open M$14.553 M$45.249
Q3 $17.742 Is Closed Keep Closed M$0 M$0
Q3 $17.742 Is Open Close Down -M$1 –M$1
Q3 $25.048 Is Closed Keep Closed M$0 M$1.187
Q3 $25.048 Is Open Keep Open M$0.042 M$1.716
Q3 $35.361 Is Closed Open Up M$7.083 M$16.7
Q3 $35.361 Is Open Keep Open M$9.083 M$18.7
Q3 $49.922 Is Closed Open Up M$19.847 M$42.228
Q3 $49.922 Is Open Keep Open M$21.847 M$44.228
Q4 $15.081 Is Closed Keep Closed M$0 M$0
Q4 $15.081 Is Open Close Down -M$1 -M$1
Q4 $21.290 Is Closed Keep Closed M$0 M$0
Q4 $21.290 Is Open Close Down -M$1 -M$1
Q4 $30.057 Is Closed Open Up M$2.433 M$2.433
Q4 $30.057 Is Open Keep Open M$4.433 M$4.433
Q4 $42.434 Is Closed Open Up M$13.282 M$13.282
Q4 $42.434 Is Open Keep Open M$15.282 M$15.282
Q4 $59.906 Is Closed Open Up M$28.599 M$28.599
Q4 $59.906 Is Open Keep Open M$30.599 M$30.599
Note: History dependent nodes are boldfaced.
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7.
Quarter PriceIncoming
StateOptimalAction R(P ) V
Q0 $28.89 Is Open Keep Open M$3.41 M$23.518
Q1 $24.556 Is Closed Keep Closed M$0 M$2.605
Q1 $24.556 Is Open Keep Open -M$0.389 M$4.167
Q1 $34.668 Is Closed Open Up M$0.475 M$29.055
Q1 $34.668 Is Open Keep Open M$8.475 M$37.055
Q2 $20.873 Is Closed Keep Closed M$0 M$0
Q2 $20.873 Is Open Close Down -M$4 -M$4
Q2 $29.468 Is Closed Open Up -M$4.084 M$5.34
Q2 $29.468 Is Open Keep Open M$3.916 M$13.34
Q2 $41.602 Is Closed Open Up M$6.553 M$37.249
Q2 $41.602 Is Open Keep Open M$14.553 M$45.249
Q3 $17.742 Is Closed Keep Closed M$0 M$0
Q3 $17.742 Is Open Close Down -M$4 -M$4
Q3 $25.048 Is Closed Keep Closed M$0 M$0
Q3 $25.048 Is Open Keep Open M$0.042 M$0.618
Q3 $35.361 Is Closed Open Up M$1.083 M$10.7
Q3 $35.361 Is Open Keep Open M$9.083 M$18.7
Q3 $49.922 Is Closed Open Up M$13.847 M$36.228
Q3 $49.922 Is Open Keep Open M$21.847 M$44.228
Q4 $15.081 Is Closed Keep Closed M$0 M$0
Q4 $15.081 Is Open Close Down -M$4 -M$4
Q4 $21.290 Is Closed Keep Closed M$0 M$0
Q4 $21.290 Is Open Keep Open -M$3.252 -M$3.252
Q4 $30.057 Is Closed Keep Closed M$0 M$0
Q4 $30.057 Is Open Keep Open M$4.433 M$4.433
Q4 $42.434 Is Closed Open Up M$7.282 M$7.282
Q4 $42.434 Is Open Keep Open M$15.282 M$15.282
Q4 $59.906 Is Closed Open Up M$22.599 M$22.599
Q4 $59.906 Is Open Keep Open M$30.599 M$30.599
Note: History dependent nodes are boldfaced.
Section A. Advanced Appendix: Monte Carlo Repeated Drawing.
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8.
Quarter PriceIncoming
StateOptimalAction R(P ) V
Q0 $28.89 Is Open Keep Open M$3.41 M$22.266
Q1 $24.556 Is Closed Keep Closed M$0 M$0
Q1 $24.556 Is Open Keep Open -M$0.389 M$1.6
Q1 $34.668 Is Closed Keep Closed M$0 M$6.463
Q1 $34.668 Is Open Keep Open M$8.475 M$37.055
Q2 $20.873 Is Closed Keep Closed M$0 M$0
Q2 $20.873 Is Open Keep Open -M$3.618 -M$9.263
Q2 $29.468 Is Closed Keep Closed M$0 M$0
Q2 $29.468 Is Open Keep Open M$3.916 M$13.34
Q2 $41.602 Is Closed Open Up -M$17.447 M$13.249
Q2 $41.602 Is Open Keep Open M$14.553 M$45.249
Q3 $17.742 Is Closed Keep Closed M$0 M$0
Q3 $17.742 Is Open Keep Open -M$6.362 -M$12.19
Q3 $25.048 Is Closed Keep Closed M$0 M$0
Q3 $25.048 Is Open Keep Open M$0.042 M$0.618
Q3 $35.361 Is Closed Keep Closed M$0 M$0
Q3 $35.361 Is Open Keep Open M$9.083 M$18.7
Q3 $49.922 Is Closed Open Up -M$10.153 M$12.228
Q3 $49.922 Is Open Keep Open M$21.847 M$44.228
Q4 $15.081 Is Closed Keep Closed M$0 M$0
Q4 $15.081 Is Open Keep Open -M$8.695 -M$8.695
Q4 $21.290 Is Closed Keep Closed M$0 M$0
Q4 $21.290 Is Open Keep Open -M$3.252 -M$3.252
Q4 $30.057 Is Closed Keep Closed M$0 M$0
Q4 $30.057 Is Open Keep Open M$4.433 M$4.433
Q4 $42.434 Is Closed Keep Closed M$0 M$0
Q4 $42.434 Is Open Keep Open M$15.282 M$15.282
Q4 $59.906 Is Closed Keep Closed M$0 M$0
Q4 $59.906 Is Open Keep Open M$30.599 M$30.599
Note: History dependent nodes are boldfaced.
9. You should have expected the costs to approach the infinite cost benchmark where you would never changethe plant state.
Cost of Opening Cost of Opening Plant Value
M$0.0 M$0.0 M$25.284
M$0.2 M$0.5 M$25.012
M$1.0 M$2.0 M$24.625
M$4.0 M$8.0 M$23.518
M$16.0 M$32.0 M$22.266
Infinite Infinite M$22.266
10. Yes, it means you fit better. Of course, if you fit 52 data points with 52 coefficients, you will fit perfectly.Unfortunately, this also means that your estimates are garbage. You want less noise, but while keeping asmall number of parameters only.
11. They fit worse. Their residuals have higher variance.
12. They must have less, because you have an extra parameter now and the AR(2) process embeds the AR(1)process. Still, the process does not fit much better, so it is not worth keeping the extra parameter.
13. Do it!
14. The former would be much tighter clustered around $28.77, while the latter would be much more variablearound $28.77.
15. It would be much tighter around $28.77.
16. The mean would be $28.89, the standard deviation would be $6.41. This would probably be a better processthan the one you just wrote down, claiming (our non-existent) managerial intuition. If you repeated the treevaluation, you would solve to find a plant value of $17.8 million. Sidenote: Below, in Table III.4, you willfind that the plant with weekly shutdown/reopening capability is worth $23.9 million. Thus, the flexibility ofweekly rather than quarterly flexibility and remaining differences in the electricity price process are togetherworth around $6 million.
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17. No. You will probably get different price scenario draws, and you did not do infinitely many so that we willagree exactly.
18. Primarily due to the electricity price process. you know this, because if you permit only quarterly changes[the other important difference from the previous subsection], you get much less in value.
19. The result is about $22.3 million, as shown in Table III.4.
20. The result is about $21.2 million, as shown in Table III.4.
21. You must cumulate undiscounted cash flows each week into your cash account, and test whether in any weekthe total sum is negative. The probabilities of this event is 7.9% for ($25,$25); 2.5% for ($20,$30); and above15% for ($15,$30).
(All answers should be treated as suspect. They have only been sketched, and not been checked.)
