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    Triantis, A.J., Real Options, in Handbook of Modern F inance, ed. D. Logue and J.

    Seward (New York: Research Institute of America), 2003, D1-D32.

    Real Options

    ALEXANDER J. TRIANTIS

    _________________________________________________________________________

    D7.01 Introduction..2

    D7.02 Shortcomings of Traditional Valuation Techniques2[1] Fundamental Principles of Valuation.3[2] Challenges in Estimating Expected Cash Flows....3[3] Challenges in Estimating Discount Rates..4[4] Linking Corporate Strategy to the Maximization of Shareholder Value...4

    D7.03 Real Options as a Conceptual Tool..5[1] Managerial Flexibility and the Option Analogy.....5[2] Taxonomy of Real Options.....6

    [a] Growth Options....6[b] Contraction Options.....7[c] Switching Options....7[d] Contractual Real Options.....8

    [3] Option-Based Strategies for Creating Value..8

    D7.04 Real Options as a Valuation Tool....9[1] An Option to Build...10[2] The Black-Scholes Formula.10[3] The Binomial Option Pricing Model....12

    [a] Generating a Binomial Tree...12[b] Using Risk-adjusted Probabilities..13[c] Accounting for Early Exercise...15

    [4] Techniques to Deal with Additional Complexities .15[a] Risk-Adjusted Decision Trees16[b] Monte-Carlo Simulation17

    D7.05 Summary19

    Suggested Reading.19

    ________________________________________________________________________

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    D7.01 INTRODUCTION

    Most corporate investment opportunities involve a sequence of managerial decisions over time. Each decision ismade based on information available at the time, but also should take into consideration future decisions under arange of scenarios. A research and development (R&D) project in a pharmaceutical company, for instance, requiresperiodic investment to be sustained. Whether additional investment will be made at a particular stage depends on theoutcome of scientific research to date, updated information on the potential size of the market for the drug, andconsideration of the flexibility to accelerate, ramp down or even cancel the project at a future date. Similarly,capacity planning for the production of semiconductor chips requires consideration of future product demandscenarios and of managerial responses, such as expansion or contraction, under these future scenarios.

    Unfortunately, the traditional implementation of discounted cash flow (DCF) or net present value (NPV) analysesdoes not properly capture the value of investments whose cash flow streams are conditional on future outcomes andmanagerial actions, and which thus have complex risk profiles. Cash flows are often estimated without explicitlyrecognizing how they depend on future investment or operational decisions. In addition, a single risk-adjusteddiscount rate is often employed, typically the companys weighted average cost of capital (WACC), and thisoverlooks changes in a projects risk over time.

    Myers (1977) first observed that future investment by corporations is discretionary, and thus is analogous to afinancial option, where an investor holds a claim to buy or sell an underlying financial asset at a potentially

    favorable price, and has the right to make this trade only if it will in fact be profitable. Myers coined the term realoptions to emphasize that investment opportunities are (or involve) options on real assets, as opposed to financialassets.

    The advantages of viewing investment opportunities as options are twofold. First, there are several well-knowninsights about options that provide us with new perspectives when evaluating investment opportunities, such as thefact that options may become more valuable if the volatility of the underlying asset increases. Second, analytictechniques have been developed for valuing options that are superior to using a standard DCF approach. Thesetechniques result in better evaluation of corporate investments with embedded options, and more accurate valuationof the securities of corporations that have such projects.

    Graham and Harvey (2001) report that 27% of the 392 CFOs they surveyed claimed to use real options always oralmost always in their capital budgeting processes. Triantis and Borison (2001) conduct in-depth interviews with

    over 35 corporate managers to better understand how real options analysis is being used by major corporations,particularly in the life sciences, energy, high tech, transportation and telecom industries. They find two different,though often complementary, approaches to implementing real options in practice: a conceptual approach, and ananalytic approach. In this chapter, I outline how these two approaches are implemented in practice, and how theycan help companies improve their strategic planning and investment evaluation processes so as to maximizeshareholder value.

    D7.02 SHORTCOMINGS OF TRADITIONAL VALUATION TECHNIQUES

    Valuation techniques are used within corporations in order to measure the effect of a project on the overall valueof the corporation, and thus of its shares. These techniques are also used by investors to assess the fundamentalvalue of financial securities, and thus to determine whether they should buy or sell these securities. Techniques such

    as DCF are viewed as being theoretically correct, and thus are routinely used in practice. However, a more criticalanalysis of the way in which these techniques are implemented suggests that the typical DCF analysis frequentlydistorts the true value of investment opportunities. To elaborate on this point, I first discuss the fundamentalprinciples of valuation, and then indicate how these may not be properly reflected in the typical implementation oftraditional valuation techniques.

    [1] Fundamental Principles of Valuation

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    The theory and practice of valuation focuses on determining how much an investor should pay today in order toreceive the right to a specific set of cash flows through time. There are two key ideas that provide guidance in howto translate a sequence of risky cash flows into a present value, or current dollar amount. First, a dollar earned inthe future for certain is worth less than a dollar today. There is (almost always) a positive real return associated withinvesting money, and there must be an additional return to compensate investors for the impact of expected inflationon the purchasing power of their money in the future. Second, investors prefer a dollar for certain rather than adollar on average, and thus risky cash flows are valued at a discount to certain cash flows. The discount depends onthe level of risk, which is assessed based on the contribution of the risky cash flow stream to the overall risk ofinvestors portfolios.

    These principles of discounting are implemented through a standard DCF computation, as follows. The average (orexpected) level of each cash flow is estimated, and then discounted back to the initial date based on the timing andrisk level of the cash flow. Discounting rates are often computed using the Capital Asset Pricing Model (CAPM),which requires the following inputs: the risk-free rate (typically the rate on a Treasury Bond with maturity consistentwith the duration of the cash flows); the risk premium required for assuming one unit of market risk (the market i susually proxied by an equity index such as the S&P 500); and the amount of market risk, which is measured by thebeta of the cash flow(s). Discounted values of each of the cash flows are then summed up to obtain the present valueof the investment opportunity.

    While theoretically sound, there are unfortunately two major problems with the typical implementation of this

    approach. First, the expected future cash flows that are estimated typically do not properly reflect the flexibility thatexists in the investment and operation of the assets producing those cash flows. Second, the cash flows at differentpoints in time typically require different discount rates to appropriately reflect their risk. Using an average orblended discount rate for all cash flows, as is typically done, may lead to significant error in the valuation of the cashflows. These two difficulties with implementing DCF in practice are discussed below in greater detail.

    [2] Challenges in Estimating Expected Cash Flows

    Too often in practice, the cash flows that are discounted in a DCF calculation reflect the most likely scenario, or theaverage scenario, rather than a properly estimated weighted average of the cash flows under all possible scenarios.As a rather simple and dramatic example of why this can seriously distort the valuation of a project, consider thedecision to purchase a peak-load generating plant that has a very high variable cost of operation. Such a plant is usedto generate electricity in rare instances when the price of electricity spikes up dramatically. Assume for simplicity

    that the plant can be started-up or shut-down quickly at no cost.

    If an analyst evaluating investment in such a peaker plant were to take the mode (most likely scenario) or themean (average) market price of electricity, and subtract the costs of generation, the resulting cash flows wouldsurely be negative, and the DCF analysis would indicate that the generating plant should not be purchased.However, the correct expected or average cash flow at each point in time must be calculated as follows. First, theanalyst should create a probability distribution of future electricity prices (e.g. by using past price data). Second, foreach possible price level, the probability of that price occurring should be multiplied by the profit (price minusvariable cost) when price exceeds cost, or zero otherwise (since the company would not operate the plant at a loss).The expected profit is the sum of these probability-weighted values over all possible price scenarios. While theprobability of large price spikes may be low, since the price, and thus profit, occasionally reach extremely highlevels, the present value of the expected profit over the life of the generating capacity may well exceed the fixed costof purchasing and maintaining the plant.

    This simple example illustrates the importance of accurately estimating future expected cash flows. In order toproperly calculate the cash flows, the future actions of management must be carefully considered. In the case of thegenerating plant, the critical decision is whether to start-up the plant each hour (or day). This decision is analogousto the exercise of a call optionif the price of electricity (the underlying asset) exceeds the cost of generation (theexercise price) during a particular hour, then the option should be exercised. The expected cash flow under allpossible electricity price scenarios is quite different than the cash flow under the expected (average) price scenario.

    In general, expected cash flows can only be properly calculated if the full range of future outcomes of key variablesis considered, together with any managerial responses over time to new information about these variables. As will

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    be shown later in Section D7.04, binomial trees and decision trees are useful approaches to visually map out thearrival of new information and the sequence of managerial decisions that take place over time.

    [3] Challenges in Estimating Discount Rates

    Once a projects expected cash flows have been carefully mapped out over the horizon of the project, these cashflows must be translated back into their present values. The rate of return on projects or companies of comparablerisk should be used as the discount rate, since this represents the opportunity cost of investing in the project.However, finding investments with truly comparable risk is extremely challenging. The risk of a project depends notonly on the volatility of underlying variables such as the price of inputs and outputs, but also on key characteristicsof the project, such as the degree of operating leverage, the timing of future investment in the project, and the extentto which management can control the cash flow profile in the future. Even small differences in these projectcharacteristics can translate into significant differences in project risk, and thus in the discount rate that should beused for valuation.

    To illustrate the difficulty in estimating discount rates, consider an R&D program that is targeted towardsdeveloping the next generation of a companys products in five years time (t=5). If we try to estimate the presentvalue at t=5 of the future cash flows that would be derived from the new products, the discount rate should reflectthe economic and market (i.e. systematic) risks of those cash flows assuming that the decision has been made tolaunch the products. Assume that this discount rate is 15%. Now, consider what discount rate should be used to

    bring the value associated with the R&D program in five years time back to the current time. Is 15% still theappropriate rate?

    In fact, the discount rate should be much higher than 15%. The investment in R&D is similar to the purchase of acall option. Some investment is made up front, but a significant investment (in production capability, marketing, andso on) must be made at a later date (in five years time) if it is optimal to exercise the option (i.e. to proceed with theproduct launch). Call options can be extremely riskythere is often a significant probability of losing all the up-front investment, but, if the option ends up in-the-money, the percentage return on the initial investment in the calloption can be very large. The same holds true for an R&D program.

    Since the risk of the R&D investment is much higher than that of the cash flows starting in five years time(assuming that the product launch decision has been made), a higher discount rate must be used during the first fiveyears of the project. How much higher than 15% this discount rate should be depends on many characteristics of the

    project, and, unfortunately, is hard to estimate accurately, as we will discuss in greater detail in Section D7.04.03. Inthe case of pharmaceutical R&D programs, Myers and Howe (1997) illustrate that discount rates are very high at theoutset of an R&D project, and then decline over the course of the project as more investment is made and as theprobability of ending up with a profitable project increases.

    [4] Linking Corporate Strategy to the Maximization of Shareholder Value

    The discussion above highlights that both the average level and the volatility of cash flows depend critically on howuncertainty is resolved over time and how managers respond to the new information. Standard DCF analyses aretypically not set up to properly map out contingent cash flows and determine appropriate discount rates. As a result,financial decision making techniques are often questioned, or even abandoned, in practice when proposedinvestments offer important strategic advantages such as flexibility, or the spawning of future growth opportunities.

    Real options analysis presents an important step in bridging the gap between finance and strategy (see Myers(1987)). It allows for the timing of information arrival and the sequence of decisions over time to be mapped out.Furthermore, it embodies the principles of finance that ensure that corporate investments are evaluated from theperspective of the companys shareholders, rather than reflecting the risk aversion of management, orinappropriately using the companys cost of capital when the projects risk differs significantly from that of thecompany. The practice of real options also typically involves careful framing of an investment decision problem inorder to improve the design of the project, and in so doing enhance shareholder value. The following section showshow real options can add value in this manner, and the subsequent section provides details on real options valuationtechniques that are used in practice.

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    D7.03 REAL OPTIONS AS A CONCEPTUAL TOOL

    Corporate decisions involve choosing from a set of alternative strategies or investments. Decisions made todaytypically affect the set of alternatives that will be available in the future. Current investments may createopportunities, or options, for future investment on possibly favorable terms. Other decisions may foreclose suchopportunities, destroying some of the companys available options. Given the important time dimension of corporatestrategy, and the significance of managements discretion to take future actions, options concepts and language havebeen increasingly adopted in strategic and operational planning. In this section, a taxonomy of different types of realoptions is provided, and the benefits of using a conceptual real options framework are presented.

    [1] Managerial Flexibility and the Option Analogy

    A defining characteristic of an option is the ability to delay making an investment decision until more information isobtained. Selecting one of investment opportunities today is not an option, but a choice between alternatives. Incontrast, having the right to choose whether to invest in a years time in one or more projects is a valuable option ifone can learn more about these investments during the next year, and if the possibility exists of investing onfavorable terms.1 Such options present themselves frequently in corporate practice given the flexibility inherent inprojects and investment programs.

    A conceptual real options approach involves identifying key sources of uncertainty and recognizing, creating, andnurturing options whose values stem from reacting to new information about the uncertainties. For instance, acompany may have an opportunity to invest in a new product. Commitment to immediate investment indevelopment, marketing and manufacturing of the product is one possibility. This alternative has the advantage ofaccelerating future cash flows and potentially preempting competition or at least securing a larger market share.

    However, these advantages may be outweighed by the option created by staging investment in the project. Newinformation can be acquired about key variables such as the size of the market, the cost of new productiontechnology, the availability of improved product designs, and the cost of key inputs. Some of this information mayarrive naturally over the course of time, while other information may be proactively obtained through investing infeasibility studies, additional R&D, and marketing surveys. The company may decide to launch the product after all,but may do so at a different scale or in a different manner that allows it to decrease costs or increase revenues, either

    for the current product, or for future generations of products.

    While the importance of flexibility in organizations has long been appreciated, the use of a conceptual real optionsframework appears to lead to superior value creation strategies that take full advantage of existing and potentialflexibilities. Many companies have found that simply using the language of options has greatly improved theintuition and communication of their managers in the strategic planning process (see Triantis and Borison (2001)).Managers improve their ability to identify a companys existing options that have resulted, for example, from anearly capture of market share. They are more likely to look out for ways to design additional flexibility into theirR&D, procurement, production, and marketing processes. They also tend to look for opportunities to purchase realoptions on favorable terms, either directly through contracts such as licenses, patents or leases, or indirectly asembedded options in procurement, sales or joint venture agreements.

    The language of real options also serves as an external communication tool when dealing with the investment

    community. Analysts have begun identifying various options, most notably growth options, that exist withincorporations. As a result, there is an increasing awareness that multiples such as price-earnings ratios may not bedirectly comparable for companies within the same industry if investment in options by some firms lead to lowershort-run earnings, followed by higher expected future earnings.

    1 In order for the option to be valuable, there must be some possibility that the investment opportunity will have apositive NPV in the future. Positive NPV opportunities arise from the existence of entry barriers, brand names,technical expertise, or other competencies that provide at least a temporary advantage over competitors. This issue isdiscussed further in Brealey and Myers (2003), Chapter 11.

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    [2] Taxonomy of Real Options

    There are a myriad of real options associated with corporate investments. It is useful to categorize these options inorder to exploit similarities between some of them, and to better appreciate their analogy to different types offinancial options. Four classes of real options - growth options, contraction options, switching options andcontractual optionsare discussed below in greater detail.

    [a] Growth Options

    Growth options are the most prevalent type of real option, and are analogous to financial call options. The cost ofthe option is the investment that is made today, or has been made at some point in the past, to create the option. Thecompany is then in a position to invest on possibly favorable terms in the future, and in so doing, to grow theirearnings. Financial call options have a limited horizon, or maturity, and some growth options also share this feature,such as oil leases (Siegel, Smith and Paddock (1987)) or options to develop real estate.

    In most cases, however, growth options have no fixed maturity date and are exercised only when the cost of waiting,due to foregone revenue or threat of competitive entry, outweigh the benefits of waiting for additional uncertaintyresolution. For instance, a consumer products company with a strong brand name may have a valuable option toexpand into international markets. It may decide to wait while additional uncertainty is resolved regarding the globaleconomic climate, foreign exchange rates, or relevant tax and regulatory policy. If the uncertainty is favorably

    resolved, the company may then invest in setting up marketing, production and distribution abroad.

    Another way in which growth options differ from standard financial call options is that most growth opportunitiesinvolve a sequence of investments over time. These growth options are thus compound options, where options arecreated upon the exercise of earlier options. Examples of such compound growth options can be found in virtuallyany industry. When a technology company decides to proceed with a new product introduction, this opens upopportunities for future related product launches or new markets. Investing in a start-up company can also be viewedas buying the first stage of a multi-stage growth option. Similarly, the decision to expand production capacity at amanufacturing company is not simply a one-time decision, since expansion decisions will impact future capacityexpansion decisions.

    In the pharmaceutical industry, R&D investment involves a series of staged-gate decisions (Schwartz (2001)). Ateach milestone, the company decides whether to exercise its option to invest in the next stage of the program.

    Exploration and development (E&D) programs in the oil and gas industry share this same feature (Smith andMcCardle (1998)). In both of these cases, market uncertainties, such as the price of a commodity or the size of aproduct market, and project specific uncertainties, such as technical uncertainty about discovery of a new drug ornew oil reserves, are resolved during each stage of the program. Market-based uncertainties are often resolvedsimply with the passage of time, thus making deferral options valuable. In contrast, companies must typically investto acquire new information about technical uncertainties. These early-stage investments, frequently referred to aslearning options, produce information rather than products, and add value by improving the quality of futuredecisions.2

    Infrastructure and platform investments are other examples of growth opportunities that companies typically havetrouble justifying without considering the options for downstream profitability that are created as a result of currentinvestment. Companies in the financial services industry, for example, have begun to evaluate informationtechnology (IT) investments by recognizing, and at times valuing, the options that are created as a result of

    implementing particular IT systems (Schwartz and Zozaya-Gorostiza (2000)). Other large-scale infrastructureinvestments such as airport expansions have also been evaluated using real options analysis (Smit (1998)).

    Finally, corporate growth often occurs through merger and acquisition (M&A) programs. Companies with activeM&A programs believe that they can acquire opportunities for future growth on favorable terms through purchasing

    2 These options can be complex in that it is not simply a matter of whether to invest in order to acquire newinformation, but also what level of investment is optimal in terms of accelerating or slowing down the informationacquisition process (see Childs and Triantis (1999)).

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    other companies. Companies such as Cisco that have a strong reputation and established product lines acquire start-ups with new technology, and in so doing expand their portfolio of growth options to develop new products overtime (Smith and Triantis (1995)).

    [b] Contraction Options

    While companies typically focus on growth initiatives, the ability to contract or abandon operations can also have asignificant impact on shareholder value. Production at chemical or refinery plants or at a mine may be scaled downwhen output prices decline, thus avoiding losses. Product lines may be abandoned or R&D and marketing effortsmay be reduced when demand declines. Of course, the degree to which a company has invested in such flexibilityup-front will affect the costs of contracting or abandoning operations.

    Abandonment and contraction options are analogous to financial put options, where an asset may be sold onpotentially favorable terms in the future. In some cases, abandonment options may have a fixed maturity date, suchas the option of a hotel operator to take down its flag from a hotel property at the end of its contract with theproperty owners. However, as with growth options, abandonment options frequently have no fixed maturity date,and may involve a sequence of decisions. Contraction may be gradual, and may allow for subsequent expansion ifconditions turn around. Even when there is a commitment to abandon a project, this may still take place in stages, inparticular when there are environmental issues that need to be dealt with over time. The decommissioning of a

    nuclear power plant, for instance, involves containment and waste management stages, and each of these stages maybe delayed over time if the benefits from potential cost reductions due to new technology outweigh the continuingmaintenance costs.

    [c] Switching Options

    While standard financial options typically offer a binary choice (to either invest or not), real options frequentlyinvolve a choice among several different alternatives. Refineries allow for a variable mix of products, as do manymodern manufacturing plants (see Triantis and Hodder, 1990). Real estate is often developed for mixed uses, andmay be redeveloped over time to adjust the mix (see Childs, Riddiough and Triantis (1996)). Many generating plantsare designed to use coal, gas or other fuels, depending on the relative prices of these inputs (see Kulatilaka (1993)).Manufacturing companies may have a global production network that allows production to be moved from one nodeof the network to another over time (Kogut and Kulatilaka (1994), and Mello, Parsons and Triantis (1995)).

    Computer hardware may be assembled from different components at various points in time based on availability,pricing, and consumer demand (see Billington, Johnson and Triantis (2002)).

    In all of these examples, flexibility has been explicitly designed into projects or production systems to allowfrequent adjustment to mixes of inputs, outputs or processes. Switching options are by nature compound optionssince there is a sequence of operational exercise decisions over time. Companies are increasingly learning how torespond to current market conditions quickly and at low cost, and thus switching options are becoming more andmore prevalent across various industries.

    [d] Contractual Real Options

    As companies develop a greater appreciation for the value of flexibility, it is more common to find flexibility tradedthrough options embedded in contracts. Firms that have invested in flexibility, or have the ability to diversify across

    customers or suppliers, are able to profitably sell flexibility (i.e. options) to parties that place a higher value on it.For example, Airbus and Boeing include aircraft purchase options in large deals negotiated with airlines. By lockingin prices and lead times for delivery, these contractual options provide airlines with value above and beyond thenatural real options they possess from being able to expand their fleets at their discretion as demand increases, butat prices and lead times that reflect current market conditions (Stonier and Triantis (1999)).

    Contraction and switching options also appear in contractual form. Aircraft manufacturers offer residual valueguarantees to airlinesthese are abandonment (put) options that provide a price floor for selling used aircraft.Contractual guarantees are also used in many other industries, and are attracting increased scrutiny given the impactof these contingent liabilities on corporate value (FASB, 2002). Contractual switching options appear in

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    procurement agreements, allowing a purchaser to select from a menu of the sellers products. For example, airlinesare often granted options (contingent rights) that allow them to select from one of a family of similarly configuredaircraft.

    [3] Option-Based Strategies for Creating Value

    While it may be useful to have a concise options-based language to refer to different types of flexibility anddownstream opportunities, the true contribution of the real options way of thinking is that companies may improvethe design of their projects or strategy. These improvements often come from well-known insights about financialoptions, extrapolated to apply to the more complex world of real options. Five key options-based strategies arepresented below.

    First, investment opportunities should not always be viewed as now ornever decisions. Clearly there are caseswhere an opportunity presents itself only at a single point in time, such as an auction for a particular asset. Also, inmany competitive situations, the risk of preemption is too high for delay to be a feasible option. In these cases, thecompany should decide immediately whether to invest or not. However, in general an opportunity should beconsidered over a period of time. While a negative NPV project should not be accepted, this decision should berevisited should market conditions change. Perhaps less intuitive is the fact that companies should not necessarilyimmediately accept a project that has a positive NPV. As is the case with in-the-money financial call options, if theunderlying uncertainty or volatility is high, the benefit of waiting to commit to an investment decision may outweigh

    any costs associated with delaying investment.

    Second, managers should not fixate on the most likely scenario and should create flexibility to allow them to alterthe course of the project. While the flexibility to expand, switch, contract or abandon may require higher up-frontcosts, it may well be justified if there is a wide range of future outcomes which makes having insurance on thedownside, or the ability to make opportunistic investments on the upside, valuable to the company. In supply chainmanagement, for instance, there has been a trend away from using point forecasts and towards using distributions ofpossible outcomes. Rather than lock in a fixed, best estimate, level of supply, tiered procurement strategies arebeing designed such that a base level of supply is locked in through forward contracts, while option contracts andspot purchases are used to cover procurement needs at higher levels.

    Third, companies should try to invest in stages rather than all at once. Each stage of investment producesinformation that can lead to better investment decisions in the future. Furthermore, the passage of time also resolves

    some of the market uncertainties surrounding a project.3

    Since time is an important value driver for options, acompanys growth options can be made more valuable by stretching out their maturity. This venture capitalistapproach of breaking up projects into stages and investing based on milestones is being increasingly adopted bymany companies.

    For example, in planning their future production capacity, Intel recently decided to adopt a staged expansionstrategy. Rather than building and fully equipping a new facility, Intel decided to only build the plant shell andthen wait for updated demand forecasts before installing expensive equipment. Intel essentially invested in a growthoption that, if exercised, could quickly provide the needed capacity, but if not, would limit their exposure undernegative economic conditions. Intel used a real options approach in order to justify investment in the first stage offacility construction (see Triantis and Borison (2001)).

    Using a real options approach, Hewlett Packard redesigned their assembly process for printers being sold into the

    European market. Originally, the printers were assembled in the U.S. based on expected demand in differentcountries. However, actual demand frequently differed from the forecasted demand, resulting in excess inventory insome countries and lost sales in others. By moving the final assembly step to Europe, printers could be configured atthe last minute for a particular market based on actual demand. The two-step assembly was more costly, but itdelayed the exercise date of the firms final assembly options, resulting in a significant net increase in value (seeBillington, Johnson and Triantis (2002)).

    3Majd and Pindyck (1987) model an investment project that requires a long time to build, and may be suspendedor abandoned as new information arrives, even though part of the project investment has already been committed.

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    Fourth, companies should try to develop a diverse set of future alternatives when planning various aspects of theircorporate strategy, including R&D, operations, and supply chain management. In R&D, it may be profitable topursue multiple, mutually exclusive, product designs. This provides a company with an option to select the mostprofitable design at a point in the future when they will have a much clearer picture of consumer tastes, productstandards, production costs and the offerings of competitors. Similarly, allowing for scope in selectingprocurement partners, distribution channels, and operational processes, while frequently more expensive thanfollowing a more focused strategy, introduces valuable chooser or switching options. The more diverse, oruncorrelated, the alternatives are, the more valuable the options become.

    Fifth, creating or purchasing real options may be quite profitable in environments where uncertainty is high. It iswell known that higher price volatility of an underlying asset increases the value of financial options on that asset(for a given current asset value). With a call option, for example, this relationship arises since the option holder cantake advantage of the upside as the asset price appreciates, but is protected against a drop in the asset price since nocommitment to purchase has been made. Similarly, a company with a large portfolio of real options effectively hasan active risk management strategy in place, and thus should not back away from volatile conditions given that itmay gain large profits on the upside, while being protected against losses on the downside.

    D7.04 REAL OPTIONS AS A VALUATION TOOL

    There are four main techniques that are utilized in practice to value real options, or projects with real optionfeatures: 1) the Black-Scholes formula; 2) the binomial option pricing model; 3) risk-adjusted decision trees; and 4)Monte-Carlo simulation.4 The Black-Scholes and binomial techniques are relatively straightforward, and areparticularly useful for problems with simple structures, such as a single source of uncertainty and a single decision.Risk-adjusted decision trees and Monte-Carlo simulation are more general and powerful techniques that can be usedin more complex settings. While the four techniques differ somewhat in terms of assumptions and structure, all ofthem reflect the same valuation principles. The four methods are illustrated and compared below in the context ofthe valuation of a growth option.

    [1] An Option to Build

    The principles of real option valuation will be illustrated using a relatively simple decision problem. On December31, 2002, a developer was considering purchasing a contractual option to build an office complex on a vacant piece

    of land. The option would expire in two yearif the developer wishes to build the property, he must do so beforethe end of 2004. We assume for simplicity of exposition that the office complex can be built instantaneously oncethe developer decides to invest. The construction cost is $95 Million at the end of 2002, but would increase eachyear at the (continuously compounded) risk-free rate of 5%, i.e. the cost would be $99.87 Million at the end of 2003or $104.99 at the end of 2004.5

    It is expected that a developed property could generate an average after-tax cash flow at a rate of $10 Million peryear at the beginning of 2003, but that the expected cash flow decreases at a (continuously compounded) rate of 2%per year. This continuous decline in cash flow generation reflects the effect of economic depreciation of the propertyas well as the impact of new office construction in the area.6 We assume an infinite time horizon, and disregard any

    4 Numerical approximation techniques, such as the finite difference approaches, for solving partial differential

    equations can also be used to value options. However, these are rarely used in practice to value real options, sincethey are less accessible to most analysts, and are often viewed as black-box approaches that lack transparencyfrom the perspective of managers making decisions.5 Continuously compounded rates of return are typically used when valuing options. The continuously compoundedrate is simply the natural logarithm of the effective return on $1 at the end of one year. For example, if $100 todayyields $105.13 risk-free at the end of one year, the annual continuously compounded rate is ln(1.0513) = .0500 (i.e.5.00%).6 One possible interpretation of the 2% annual decay rate in net cash flows is that investment of (g+2)% of the valueof the property is required on an ongoing basis to maintain the property in a condition whereby operating cash flowscan grow at the rate of g%.

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    future managerial actions (such as renovations, expansions, etc.) that may impact the cash flows. In other words,there is only a single decision, which is whether to develop the property or not.

    Assume that, based on market returns for comparable properties in the same geographic region, a (continuouslycompounded) capitalization rate of 8% should be used to discount future cash flows. Given the simple structure ofthis problem, the developed property can be valued as a declining perpetuity.7 The value at the initial decision pointis simply $10 / (.08 + .02) = $100 Million. The value at the end of the 2003, assuming the expected case of a 2%decline in the cash flow, would be $10 e-.02 / (.08 + .02) = $98.02, and this value will continue to decline by 2% peryear assuming the expected decline in cash flows materializes and the capitalization rate stays constant (whichimplies the level of risk, the risk premium, and the risk-free rate all remain constant).

    [2] The Black-Scholes Formula

    The Black-Scholes formula is a well-known formula for valuing financial options, principally because it was thefirst, and is the simplest, formula available for pricing options with finite maturities. The advantage of the Black-Scholes model relative to other valuation techniques is that it simply requires plugging in six inputs into a formula.While this is an attractive feature, there are some important limitations to the Black-Scholes formula. First, itassumes that options can be exercised only at their maturity date. Second, it assumes that the underlying asset valuehas a lognormal distribution, which implies that the (continuously compounded) rate of return on the underlyingasset price is normally distributed with a constant standard deviation over time. Third, the formula lacks in

    transparency and intuition, and thus has the disadvantage of coming across as a black-box. These limitations willbe revisited below, but first I show how the formula can be used to value the option to build.

    The six inputs that are required are as follows: 1) the initial value (S) of the underlying asset, which in the exampleis the initial value of the developed property ($100 Million); 2) the time (T) until maturity of the option (2 years); 3)the investment, or exercise price (X), that is required to exercise the option ($104.99 Million at the end of 2004); 4)

    the difference ( ) between the capitalization rate and the percentage expected change in the value of the underlyingasset (10% = 8% - (-2%)), which is equivalent to the percentage annual cash return on the property8; 5) the

    continuously compounded annual risk-free rate of return (r=5%); and 6) the volatility ( ) of the underlying asset,which is the standard deviation of the rate of return on the underlying asset, which for the developed property, willbe assumed equal to 20%.9

    Why do these parameters affect the value of a real option and how are they captured within the Black-Scholes

    model? The value of the development option depends on the likelihood that the property will in fact be developed,and on how profitable such development might be. As volatility increases, there is a larger probability thatdevelopment will be very profitable, thus increasing the value of the option.10 A longer horizon also helps the option

    become deeper in-the-money. The initial property value, the risk-free return and the parameter must also bereflected in the option value since they together determine the expected value of the developed property at the

    maturity date: S and r should enter into the valuation formula with a positive sign, and with a negative sign.Finally, the value of the option should be inversely related to the cost of exercising it, i.e. the cost of developing theproperty.

    7 The valuation formula for a growing perpetuity is CF1 / (rg), where CF1 is the cash flow at the end of the firstperiod, r is the annual capitalization (discount) rate, and g is the annual growth rate (which in the example aboveis negative). When the cash flow is continuous, as in the development example, CF1 is the initial annual cash flow

    generation rate.8 In the case of a financial option, this parameter is the percentage payout or yield on the underlying asset, such as adividend yield on a stock. The dividend yield is simply the difference between the total expected rate of return(capitalization rate) on the stock and the expected percentage change in the present value, or price, of the stock.9 Given that we are assuming that the rate of return on the developed property is normally distributed and that theexpected percentage change in the value of the property is2%, a volatility of 20% implies that there is roughly a68% chance that the annual change in value will be between22% and +18%, a 16% probability of a capital loss ofgreater than 22%, and a 16% change of a capital gain of greater than +18%.10 The probability of a large drop in the value of a developed property also increases, but development would notoccur under these circumstances, so this is not material.

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    The Black-Scholes formula for the value of a call option is as follows11:

    )()( 21 dNXedNSeCrTT

    TddandTTrXSdwhere 122

    1 ,/])5.()/[ln(

    N(d) is the cumulative standard normal distribution function (the Excel function NORMSDIST computes thisvalue). It is the probability that a normally distributed variable with a mean of zero and a standard deviation of onewould have a value less than d. ln denotes the natural logarithm.

    It is difficult to develop sound intuition regarding the Black-Scholes model. The first main term reflects the presentvalue of the developed property, conditional on it being developed. The second term reflects the cost of thedevelopment, discounted to the present date, and multiplied by the risk-adjusted probability, N(d2), that thedevelopment cost will be incurred, i.e. that development will actually take place at the end of 2004. Note that risk isaccounted for in the probability of development, rather than the discount rate. This important issue will be discussedin greater detail in the context of the binomial option pricing model, where it is somewhat more transparent.

    Applying the Black-Scholes formula to the development option, we substitute the following input values into the

    equation: S=100, X=104.99, T=2, =.10, =.20, and r=.05.

    350.0)(,384.022./]2))2(.5.10.05(.)99.104/100[ln( 12

    1 dNd

    252.0)(,667.022.384.0 22 dNd

    71.4)252(.99.104)350(.100 2)05(.2)10(. eeC

    To help the reader develop better intuition about option values, sensitivity analysis on two key value drivers,

    (volatility) and (delta), is provided in Table D7-1. Option values are quite sensitive to these two parameters: even asmall increase in volatility can significantly raise the value of an option; similarly, as the loss from waiting, ascaptured by delta, decreases, the option becomes significantly more valuable. Note that it is conventional in asensitivity analysis to keep all other input variables constant. Keep in mind, however, that an increase in volatilitymay simultaneously decrease the property value if the additional risk is systematic (correlated with the market).Similarly, an increase in delta will lead to a lower property value unless the initial cash flow is simultaneouslyincreased.

    [Table D7-1 here]

    Recall that the Black-Scholes value is based on the assumption that development (exercise of the option) can onlytake place at the end of 2004. But, note that the option value of $4.71 Million calculated by the Black-Scholesformula is lower than the value of immediate development ($5 Million = 100-95)! . In this example, the developergives up significant cash flow by waiting to develop, and the value of waiting for two years to get more informationabout the ultimate property value does not offset this loss in cash flow. However, while immediate development maybe preferable to waiting for two years, there may be a superior, though somewhat complex, solution to this decisionproblem. Perhaps the developer should wait for one year to get more information: if the property value increasesduring this period, the developer may commit to developing at the end of 2003; if the value does not increase, thedeveloper may wait for another year before making a commitment.

    11 An alternative to plugging numbers into the Black-Scholes formula is to use a look-up table that is based on the

    formula. It is relatively straightforward to re-express the formula in terms of only two variables: T and Se(r- )T/X.By calculating the values of these two terms and looking up the corresponding row and column in the table, theoption value can be determined (interpolation between values found in the table is typically required, and there isalso a final multiplication step that is involved). Luehrman (1998) provides a Black-Scholes look-up table, andillustrates the procedure to value a growth option.

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    Unfortunately, there is no easy way to adapt the Black-Scholes model to examine such complex investmentstrategies, and thus to capture the full value of the option. Rather, an alternative technique is required that cansystematically value the option through time. The binomial option pricing model is such a technique. It also has theadvantage of transparency relative to the Black-Scholes formula, and thus leads to better intuition regarding wherethe value of an option comes from.

    [3] The Binomial Option Pricing Model

    While the Black-Scholes model assumes that the value of the underlying asset is lognormally distributed, thebinomial model assumes that in each time period the value of the underlying asset can take on only one of twovalues. By allowing for a sequence of periods with such binomial movements, a large set of paths (a binomialtree) can be generated that closely approximates all possible value changes that could occur to the underlying assetduring the life of the option. Based on the values generated for the underlying asset, and the probabilities ofachieving these values, the expected payoff to the option can be calculated, thus leading to a valuation of the option.The steps of the binomial method and the treatment of risk in the valuation procedure are described below, andillustrated using the development option example.

    [a] Generating a Binomial Tree

    A standard way of setting up a binomial tree is to assume that during each time period of length T the value of the

    underlying asset either increases by a factor of u=e(r- ) T, or decreases by a factor of d=e(r- )- T. These returnfactors are chosen to represent two points that are a standard deviation above or below the forward value for theproperty, i.e. a value that could (at least conceptually) be locked in at the present time for purchasing a developed

    property in T years.12 Using the parameter values from the development option example, and assuming for

    simplicity that the two-year horizon of the problem is split into two time periods, each with duration ( T) of oneyear, u=1.1618 and d=0.7788.

    Starting with the initial value of the developed property of $100 Million, and multiplying by the return factors u andd, the two possible values of the developed property at the end of the first year (t=1) are 116.18 and 77.88, asillustrated in Figure D7-2. The tree generation procedure continues by multiplying each of these two values by u andd in order to obtain the values for the end of the second year (t=2). Note that there are only three, rather than four,values (134.99, 90.48, 60.65) at the end of the second year since it is irrelevant whether the underlying asset valuegoes down first and then up, or vice-versa. In other words, the branches of the tree recombine, which is an

    attractive computational feature of such trees.13 If the two-year horizon of the decision problem were broken downinto 24 monthly periods, rather than just two yearly periods, there would be 25 possible values for the underlyingasset at the end of the second year, providing a closer approximation to the full distribution of possible values.

    [Figure D7-2 here]

    Once the values of the underlying asset have been generated, the next step is to calculate the value of the option atthe maturity date under each of the value scenarios. This simply requires subtracting the exercise price of 104.99(i.e. the cost of development) from the developed property value at the end of the two year horizon if the option isin-the-money (i.e. when the property value is 134.99, the option value at the maturity date is 29.99), or else settingthe option value equal to zero if the option is out-of-the-money, i.e. if the property should not be developed.

    [b] Using Risk-Adjusted Probabilities

    The next step requires working backwards through the tree to first calculate the option value at the end of the firstyear (t=1) under each of the two possible value scenarios, and then to calculate the option value at the initial date

    12 There are alternative ways to generate the up and down return factors which lead to the same option valuationsolution as long as there are a sufficient number of periods in the binomial tree (see McDonald, (2003)).13 The ability to create a recombining tree is important in that it greatly reduces the number of computations fortrees with many periods. It is easy to see that this feature will occur if the up and down returns are constant throughtime. However, even if volatility, and thus the up and down returns, are time varying, there exist ways to createrecombining trees.

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    (t=0). These option values can be calculated using a DCF-type technique, but with a clever twist. In a conventionalDCF calculation, the expected cash flow is estimated, and then discounted at a rate that reflects the riskiness of thecash flows. In contrast, in the binomial model, the expected value of the option is adjusted to account for theoptions risk, and is then discounted at a risk-free rate.

    To see how, and why, this is done, we start by looking more carefully at the first year of the binomial tree. The valueof the developed property at the initial date is $100 Million, which is the present value of all the future expectedcash flows obtained from the property. This present value could alternatively be broken down into the present value

    of the cash flows during the first year, which can be shown to equal 100(1-e- ), or $9.52 Million, and the present

    value of the expected property value at the end of the first year, which is 100e- , or $90.48 Million. The latter

    implies that the expected property value at t=1 must be (100e- )e.08, or 100e-.02 = $98.02 Million (since the discountrate for the property is 8%). Given the binomial model structure with a value of $116.18 Million in the up state and$77.88 Million in the down state at t=1, the probability (p) of the property value increasing must be approximately53% (98.2 = (.53) 116.18 + (1-.53) 77.88).

    Now, consider that, while the typical practice in DCF analysis is to account for risk by adjusting the discount rate,there is nothing in financial theory that requires that the denominator, rather than the numerator, incorporate the riskadjustment. Risk can be accounted for in the numerator by decreasing the expected cash flow to reflect a penalty ordiscount for risk, and the same valuation should be obtained as if the discount rate were risk-adjusted.

    A conceptually appealing way of understanding this procedure is that there is a risk-free amount, called theCertainty Equivalent Value (CEV), which would be less than the expected value of the cash flow, but that investorswould view as equivalent to the risky cash flow. In calculating the present value, the CEV should be discounted at arisk-free rate, since it is indeed risk-free. Thus, risk is accounted for in the numerator rather than the denominator. Inthe property development example, the CEV at t=1 must be $95.12, since 95.12e -.05 = $90.48, the present value att=0 of the t=1 property value (note that the risk-free rate of 5% is used for discounting).

    The numerator could alternatively be adjusted for risk as follows. Risk-adjusted probabilities (often referred to asrisk-neutral probabilities) could be derived, such that, when they are multiplied by the value of the underlying assetin the up and down nodes at t=1, a value of $95.12 Million (which is equivalent to the CEV) is obtained. The risk-adjusted probability, p*, of an increase in the property value during the first year must solve the equation p*(116.18) + (1-p*) (77.88) = 95.12. The risk-adjusted probability, p* = 45%, is less than the actual probability of avalue increase (p=53%) calculated earlier. This suggests that the expected cash flow is being penalized or discounted

    for risk by reducing the probability of a value increase, while increasing the probability of a value drop.

    The point of deriving these risk-adjusted probabilities is that they allow for a simple way to value options. As will beshown later, the risk level of options can be very difficult to estimate accurately, and thus risk-adjusting discountrates in a binomial tree can be extremely challenging and prone to error. Instead of following the traditional DCFapproach, risk-adjusted probabilities can be used to calculate a risk-adjusted expected value of the option, and then arisk-free rate can be used for discounting.

    In the context of the development option, the risk-adjusted probabilities of an increase and decrease in the value ofthe developed property were found to be 45% and 55%, respectively. While these probabilities were calculated forthe first year of the tree, they also apply to the second year of the tree.14 Starting at t=1, the value of the option whenS=$116.18 Million can be calculated as the risk-adjusted expected value of the option at the end of the second year,discounted back to t=1 at the risk-free rate. This option value is (.45 * 29.99 + .55 * 0) e-.05 = $12.84 Million (see

    Figure D7-2). The option value at t=1 when S = $77.88 Million is simply zero since the option values for thecorresponding up and down nodes at t=2 are both zero. Going backwards in time to the initial date, the option valueis calculated as (.45 * 12.84 + .55 * 0) e-.05 = $5.50 Million.

    While this value differs from the Black-Scholes value calculated in the previous section ($4.71 Million), if thenumber of periods in the binomial tree were to increase (e.g. from annual periods to weekly periods), the value fromthe binomial model would quickly approach the Black-Scholes value. This occurs since, with a large number of up

    14 As long as , , and r do not vary over time, the risk-adjusted probabilities will be the same in each period of thetree.

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    or down moves, the distribution of the value of the underlying asset at the maturity date approximates the lognormaldistribution which the Black-Scholes model assumes (see McDonald (2003), Section 11.3).

    It is natural at this point for the reader who is seeing risk-adjusted probabilities for the first time to be uncomfortablewith the valuation method that has been presented. Why does this technique work? While a formal demonstration ofthe validity of this technique is not difficult, it is beyond the scope of this chapter (see McDonald (2003), Appendix11B, for a nice exposition of this material).15 Yet, some intuition can be obtained simply by understanding thatchanges in the value of an option are driven directly by changes in the value of the underlying asset. The changes inthe underlying asset are represented by the sequence of up and down moves through a binomial tree, and the riskrepresented by these movements in the underlying asset can be captured through risk-adjusting the correspondingprobabilities. In other words, since the risk of an option is driven by the risk of its underlying asset, if we can capturethe asset risk in the risk-adjusted probabilities, these can be directly used to adjust for risk when valuing the option.

    As mentioned earlier, trying to value an option by using a traditional DCF technique with risk-adjusted discountrates is too challenging to execute properly. To illustrate this important point, let us back out what risk-adjusteddiscount rates would be necessary to arrive at the option valuation shown in Figure D7-2. At t=0, the option value is$5.50 Million, which is the discounted value of the expected option value at t=1. Given the actual (true)probabilities, p = 53% and 1-p = 47%, the risk-adjusted discount rate must be 21% to obtain the option value of$5.50 Million at t=0 ($5.50 = (.53 * 12.84 + .47 * 0) e-.21). This rate is significantly higher than the 8% discount ratefor the property value! In general, not only will discount rates be higher than the capitalization rate for the

    underlying asset, they will also change over time and will depend on the value of the underlying asset relative to theexercise price (see McDonald (2003), Table 19.1). As a result, there is no simple general way to obtain appropriaterisk-adjusted discount rates, and thus the risk-adjusted probability methodology is extremely useful when valuingoptions.

    [c] Accounting for Early Exercise

    The valuation of the development option in the previous section implicitly assumed that the option would not beexercised before the end of the two-year period. Yet, given that a significant cash flow is foregone by waiting todevelop, it is quite possible that, under certain circumstances, the option should be exercised before the optionsmaturity date. Fortunately, this ability to exercise early can be easily accommodated within the binomial optionvaluation model, an important advantage relative to the Black-Scholes formula.

    Using the two-period binomial model constructed in the previous section, we must check for early-exercise at twopoints: at t=0 (i.e. immediate development) and at t=1 (develop in one years time). In the high value scenario at t=1(S=$116.18 Million), the option value was calculated to be $12.84 Million in the last section (Figure D7-2).However, this option value, which considers the value of the option at the end of the second year and thusincorporates the benefit of waiting for more information, is lower than the profit gained from developing theproperty at the end of the first year, which is $16.31 Million ($116.18 Million minus the cost of development at t=1,$99.87 Million). The value of the option in this scenario is thus $16.31 Million, reflecting the optimal decision todevelop (see Figure D7-3). For the lower value scenario at the end of the first year, the option to develop is out-of-the-money, and thus the developer would hold on to the option for another year (though, in this example, the optionhas zero value).

    [Figure D7-3 here]

    Stepping back to the initial date (t=0), the property could either be developed immediately or held on to for at leastone year. Immediate development has an NPV of $5 Million. The value of the option to defer development is simplythe present value of the expected option value at t=1, were the option value when S=$116.18 Million at t=1 now

    15 Under some commonly made assumptions, one can construct a portfolio consisting of a specific asset and risk-free bonds, which, if properly rebalanced over time, can exactly replicate an option on the asset. The options valueshould thus track the value of the portfolio of the asset and the bond. This replication argument is at the core ofoption pricing theory, and can be shown to be consistent with the valuation methodology that is based on risk-adjusted probabilities. It can also be used to provide a formal relationship between the risk of an option and that ofthe underlying asset.

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    reflects the optimal exercise decision in that scenario. This value of $6.99 Million (= (.45 * 16.31 + 0) e-.05), is largerthan the value of immediate development, and thus the developer should delay development for one year. Note thatthe option value at t=1 is higher than that computed in the previous section ($5.50 Million). The difference betweenthese two option values represents the value of being able to exercise the option early. In general, early exercise islikely to occur if the option can become deep-in-the-money before the maturity date, if the yield on the underlyingasset is large, and if the volatility is relatively low. When these conditions are presentand they often are in realoptions problems - using the Black-Scholes formula will understate the true value of the option.

    [4] Techniques to Deal with Additional Complexities

    The Black-Scholes and binomial option valuation models are widely used in practice for valuing growth options,such as the development option illustrated above, as well as many other types of real options. However, severalsimplifying assumptions are often required to make the decision problems fit these models. For instance, in thedevelopment option example, four key assumptions were made: there was only one source of uncertainty, thisuncertainty had a lognormal value distribution, there was a single decision to be made (develop or not), and therewas a finite horizon of two years.

    While such simplifying assumptions may be relatively innocuous for some decision or valuation problems, they maylead to significantly distorted valuations and suboptimal investment decisions in many other cases. Fortunately,there are two more general real options valuation techniquesMonte Carlo simulation and risk-adjusted decision

    trees - that can be used for more complex decision problems. These techniques, and their relative strengths, areoutlined briefly below.

    [a] Risk-Adjusted Decision Trees

    Decision trees have long been used to model decision problems by mapping out a sequence of information arrival(event) and decision points over time. While binomial trees were later independently developed to address optionvaluation problems, the two types of trees have some natural similarities in the way optimal decisions are derived bystarting at the end of the horizon and working back through time (see Smith and Nau (1985)).16

    Binomial option valuation trees provide two advantages relative to traditional decision tree analysis. First, abinomial tree is somewhat simpler to lay out and to look at since the decision node is implicit at each event node inthe tree. More importantly, binomial option valuation trees properly account for risk from the perspective of a

    companys shareholders. A standard decision tree analysis of the property development problem would have used acost of capital for the developer (most likely the 8% capitalization rate), rather than reflecting the higher risk of thedevelopment option in the valuation.

    However, the technique used to adjust for risk in binomial trees can be directly translated into a decision treeanalysis framework. Furthermore, decision trees have the advantage of being better able to map out more complexproblems that have several uncertainties that are resolved at different rates, and sequences of decisions to be madeover time. Thus, a more general form of the binomial option valuation tree technique is a risk-adjusted decisiontree. Depending on ones perspective, this technique can be viewed either as a decision tree that is properly alignedwith the objective of maximizing shareholder wealth, or as an expanded binomial option valuation tree which allowsfor several uncertainties and decisions.

    To illustrate the use of a risk-adjusted decision tree when there is more than one uncertainty and there is a sequence

    of decisions, consider the property development option from the previous section, but with the following twoadditional complexities. First, the cost of development depends on technical and regulatory uncertainties that will beresolved at the end of the first year. The actual cost of development will be either 20% higher or 20% lower, withequal probability, than the expected costs of $99.87 at t=1 or 104.99 at t=2. Second, if the property is developed att=1, the property owners will be permitted to expand the capacity, and thus the value, of their property by 50% att=2. The cost of expansion will be half of the cost of development at t=2 (since the extra capacity is half of the initialcapacity), and the developer may decide not to expand the property if the cost of expansion exceeds the additionalvalue generated.

    16This solution approach is an application of an optimization technique termed dynamic programming.

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    A decision tree for this more complex version of the development problem is shown in Figure D7-4 (the completetree has been pruned somewhat for the sake of simplicity by removing decision paths that will clearly not befollowed). The circles denote event nodes, or points where new information is obtained about the property value orthe cost of development (the values for S and X are shown for each branch emanating from the event node, e.g.116.18 for S Up at t=1). The squares denote decision points, i.e. whether to develop or delay at t=1, and whether toexpand or stay (i.e. not expand) at t=2 if development took place at t=1, or whether to develop or abandon at t=2 ifdevelopment hasnt already taken place.

    [Figure D7-4 here]

    In order to solve the decision tree, risk-adjusted probabilities are needed to calculate expected values at t=2, whichare discounted back to t=1 at the risk-free rate, and similarly to calculate the option value at t=0 based on optionvalues at t=1. These risk-adjusted probabilities for the uncertainties are indicated along the up and down branchesemanating from the event nodes. For the property value, the risk-adjusted probabilities are the same as shown inSection D7.04.3b. For the cost of development, since the technical and regulatory risk driving the cost uncertainty ispresumably risk that shareholders can diversify, the probabilities do not need to be adjusted for risk.

    The values corresponding to the optimal decisions are shown in Figure D7-4 to the immediate left of each decisionnode. For instance, consider the option to expand at t=2 if the property value has increased in both years and the

    development cost is at the 80% level ($83.99 Million = $104.99 * .80). The value of the expanded property ($134.99* 1.50) minus the cost of expansion ($83.99 * .5) yields a payoff at t=2 of $160.48 Million (the cost of the initialdevelopment will be subtracted out when calculating the value at t=1). The owner should expand the property giventhat this value of $160.48 Million exceeds that of the Stay alternative, which is simply the value of the property att=2, $134.99 Million. Going back in time to t=1, the value associated with the Develop node is $47.20. This iscalculated by discounting the expected value of the option at t=2, (.45 * 160.48 + .55 * 90.48) e -.05, subtracting thecost of development at t=1, 79.90, and adding the present value of the cash flow received during the second year,$116.18 (1-e-.10).

    The optimal development policy is as follows. If the property value increases during the first year and the cost ofdevelopment ends up at the 80% level, development should occur. In all other cases, development should bedelayed. If development has not already occurred during the first year, it should take place at the end of the secondyear if the property value increases in both years and the development cost is high, or if the property value increases

    in one of the two years and the development cost is low. If the property value increases in both years and thedevelopment cost is low, the developed property should be expanded at the end of the second year.

    The option value of $11.66 Million shown at the left of Figure D7-4 is higher than in the example in sectionD7.04.3c (where the option value was $6.99 Million). This is due to two factors. First, introducing an additionallevel of uncertainty (the cost of development) increases the overall volatility underlying the option, and thusincreases the value of the option. Second, there is now an additional option to expand capacity, which turns out to bequite valuable in the scenario where the property value continues to increase over time.

    The example in this section is meant as a relatively simple illustration of how a decision tree can be set up andsolved when there is more than one uncertainty and when there is a sequence of interrelated decisions. It is possibleto reconfigure a binomial model in ways so as to accommodate these complexities, but typically the problem is mosteasily set up and viewed as a decision tree. Software packages such as DPL or Precision Tree can be used to value

    the option or project depicted in the tree, as well as to determine the optimal investment or operating policies.

    17

    Risk-adjusted decision trees have thus been used in practice to solve problems involving a series of decision pointsand multiple uncertainties, such as the evaluation of R&D programs.

    17 Decision trees frequently use three branches, rather than two, to model uncertainties. These three branches maycorrespond, for instance, to the 10-50-90 percentile points from the uncertaintys distribution. Trinomial lattices ortrees have also been used to value options, using the same general procedure as for the binomial model. In somecases, such as when the underlying uncertainty exhibits mean reversion, trinomial models may be computationallymore appropriate.

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    [b] Monte-Carlo Simulation

    Monte-Carlo simulation draws on the same basic valuation principles as the other real option valuation techniquespresented above. First, a number of different values for the underlying uncertainties are generated based ondistributions (probabilities) that are adjusted for systematic risk. The expected value of the option is then calculated,and a risk-free rate is used to discount this expected value back to the initial date. The principal advantage thatMonte-Carlo simulation provides over the other valuation techniques is its ability to deal with multiple uncertainties,in particular if they have non-standard distributions and may be correlated with each other. For instance, Monte-Carlo simulation would be the method of choice to evaluate the peak-load generating plant mentioned earlier in thechapter, since electricity prices are subject to occasional spikes which are most easily generated through asimulation.

    As a simple illustration of this valuation methodology, consider the development option problem where the propertyvalue at t=2 is lognormally distributed with the parameters specified earlier, and where the cost of development willbe either $83.99 Million (80% of 104.99) or $125.99 Million (120% of 104.99) at t=2, with equal probability.Assume for simplicity that the option can be exercised only at the end of the two-year period. A Monte-Carlosimulation can be easily implemented using an Excel add-in package such as Crystal Ball, as follows.18

    First, assign spreadsheet cells for the property value (S) and the development cost (X) at t=2. Then, select theappropriate distribution for each variable from a menu provided by Crystal Ball, and enter the relevant parameters

    for the distribution. The property value distribution is generated by setting S=100 e Y, where Y is normallydistributed with a mean of (r- - .5 2)*T =-0.14, and a standard deviation of T = .2828.19 This ensures that thevalue of the property at t=2, discounted back for two years at the risk-free rate, will indeed be $100 Million.

    In an output cell, enter the discounted value of the option payoff, i.e. Max(S-X,0) * e-.05 *2. Click on the button thatstarts the simulation. Foreach run, values are randomly selected from the distributions of S and X. Figure D7 -5displays the distribution of (S-X) based on 100,000 runs of the simulation. This value is negative x% of the time,and development would thus not occur in those cases. Once the simulation procedure is completed, the average ofthe discounted option values obtained from all the runs is reported, which is $6.86 Million for the developmentoption.20

    [Figure D7-5 here]

    This development option example is a simple illustration of how Monte Carlo simulation can be used to value a realoption. It does not demonstrate the full capability of this valuation technique. For example, rather than generating aprobability distribution for the property value, one could start with distributions for each of the yearly cash flows,and then derive the property value distribution. This is particularly useful in cases where information about the valueof a real asset may not be easy to obtain, but corporate data on cash flows exist and a value model can be built basedon this data.

    Monte-Carlo simulation is also useful for problems that exhibit a property called path dependency, where futuredecisions or outcomes depend on decisions made at earlier points in time. For example, consider the valuation of thefollowing guarantee made by a hotel operator to the owner of the hotel property. The hotel operator promises tomake up any shortfall in the owners profits below $10 Million each year for the next five years. However, there is amaximum payout of $25 Million over the life of the guarantee, and if the owners profit exceeds the $10 Millionlevel in any year, the surplus can be used to reduce the guarantee payout from previous years. These two conditions

    18 Mun (2003) provides details on using Crystal Ball software to value real options.19 See McDonald (2003), Chapter 18, for details on the transformation between the normal and lognormaldistributions, and how to risk-adjust these distributions.20 Since the cost of development is a diversifiable source of risk, and since the property value is lognormallydistributed, one can actually value this option in a relatively straightforward manner using the Black-Scholesformula. Calculate the Black-Scholes values with an exercise price of $83.99 (B-S value = 12.12) and $125.99 (B-Svalue = 1.62). Since there is a 50/50 probability of the high or low development cost, take the simple average of thetwo Black-Scholes values to obtain $6.87 Million. Note, however, that this simple trick does not work if thedevelopment cost is correlated with the property value.

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    create path-dependency in the problem, since the option payout in a given year depends on payouts in prior years. Itis generally quite challenging to build trees that can account for such path-dependencies, and simulation is thenatural technique to address such option valuation problems.

    Given the power of Monte-Carlo simulation, it may seem surprising that it is not used even more frequently inpractice. There are two explanations for this lack of widespread acceptance. First, the solution procedure may appearless transparent to management than a binomial or decision tree model, where one can trace along the few paths thatare generated. However, new software packages with good graphical interfaces seem to be addressing this concern.

    Second, and more importantly, standard Monte-Carlo simulation techniques have been unable to solve optionproblems where early exercise is involved, or more generally where decisions at a point in time depend on decisionsmade at a future point in time (i.e. dynamic programming problems). However, techniques have recently beendeveloped to adapt Monte-Carlo simulation to solve such problems (see Longstaff and Schwartz (2001) and Broadieand Glasserman (1997)). Schwartz (2001) provides an interesting application of these techniques to the evaluation ofpharmaceutical R&D projects. The use of Monte-Carlo simulation to value real options is likely to grow over timegiven its power and the development of software packages to support such applications.

    D7.05 SUMMARY

    Real options analysis addresses many of the shortcomings of conventional financial decision making techniquescurrently in use. Real options explicitly considers how management will react to a broad range of scenarios overtime, and appropriately captures the value of flexibility and future growth opportunities associated with a project.Companies appear to be benefiting from using real options even if only as a conceptual framework for evaluatinginvestment and operating strategies. Companies that implement real options techniques to value investmentopportunities are able to more precisely evaluate the impact of corporate projects on shareholder value. Furthermore,they are able to use the contingent strategies that are generated as part of a real options analysis to better manageaccepted projects.

    Based on survey data collected at various points during the last three decades, corporate adoption of the NPVtechnique has been rather gradual (see Table 1 in Jagannathan and Meier (2002)). It is perhaps not surprising that,while corporate managers have quickly adopted a real options way of thinking, the implementation of rigorous realoptions valuation techniques has taken place at a slower pace. Many companies, particularly those in more science

    or engineering related industries, have increasingly applied real options techniques with positive results. As the buy-in from senior management increases and in-house expertise is developed, real options analysis will graduallybecome an integral component of corporate decision making. At the same time, the investor community willincreasingly appreciate the limitations of conventional DCF analysis and the need to apply more appropriateoptions-based techniques for corporate and equity valuation.

    Suggested Reading

    Amram, M. and N. Kulatilaka,Real Options: Managing Strategic Investment in an Uncertain World. Boston:Harvard Business School Press, 1999.

    Billington, C., B. Johnson and A.J. Triantis, A Real Options Perspective on Supply Chain Management in HighTech,Journal of Applied Corporate Finance, Vol. 15 (2), Summer 2002.

    Brealey, R.A. and S.C. Myers.Principles of Corporate Finance, 7th ed. New York: McGraw Hill, 2003.

    Brennan, Michael J. and Eduardo S. Schwartz, Evaluating Natural Resource Investments,Journal of Business,Vol. 58 (1985), pp. 135-157.

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    Broadie, M. and P. Glasserman, Pricing American Style Securities by Simulation,Journal of Economic Dynamicsand Control, Vol. 21 (1997), 1323-1352.

    Childs, P., S. Ott, and A.J.Triantis, Capital Budgeting for Interrelated Projects: A Real Options Approach, Journalof Financial and Quantitative Analysis, Vol. 33, No. 3, September 1998, 305-334.

    Childs, P., T. Riddiough, and A. J. Triantis, Mixed-Uses and the Redevelopment Option,Real Estate Economics,Vol. 24, No. 3, 1996, 317-339.

    Childs, P. and A. J. Triantis, Dynamic R&D Investment Policies,Management Science, Vol. 45, No. 10, October1999, 1359-1377.

    Copeland, T. and V. Antikarov.Real Options: A Practitioners Guide. New York: Texere, 2002.

    Dixit, A. and R. Pindyck. The Options Approach to Capital Investment, Harvard Business Review, Vol. 73,(May/June 1995), pp. 105-115.

    FASB (Financial Accounting Standards Board), Guarantors Accounting and Disclosure Requirements forGuarantees, Including Indirect Guarantees of Indebtedness of Others: Exposure Draft of Proposed Interpretation ofFASB Statements No. 5, 57, and 107 (May 22, 2002).

    Graham, J. and C. Harvey, The Theory and Practice of Corporate Finance: Evidence from the Field Journal ofFinancial Economics, Vol. 60 (2001), 187-243.

    Grenadier, S. (editor), Game Choices: The Intersection of Real Options and Game Theory, London: Risk Books,2000.

    Ingersoll, J. and S. Ross. Waiting to Invest: Investment and Uncertainty,Journal of Business, Vol. 65, No. 1,(1992), 1-29.

    Jagannathan, R. and I. Meier, Do we Need CAPM for Capital Budgeting? Financial Management, Vol. 31, No. 4(2002), 55-77.

    Kester, W. C., Todays Options for Tomorrows Growth,Harvard Business Review, (1984), pp. 153-160.

    Kogut, B. and N. Kulatilaka, Operating Flexibility, Global Manufacturing, and the Option Value of a MultinationalNetwork,Management Science, Vol. 40 (1994), 123-139.

    Kulatilaka, N. The Value of Flexibility: The Case of a Dual-Fuel Industrial Steam Boiler,Financial Management,Vol. 22, (1993), pp. 271-280.

    Longstaff, F.A. and E.S. Schwartz, "Valuing American Options by Simulation: A Simple Least-Square Approach",Review of Financial Studies, Vol. 14(1) (Spring 2001), 113-147.

    Luehrman, T. Investment Opportunities as Real Options: Getting Started With the Numbers, Harvard BusinessReview, July-August 1998, 51-64.

    Majd, S. and R. Pindyck Time to Build, Option Value, and Investment Decisions, Journal of FinancialEconomics, Vol. 18 (1) (1987), 7-27.

    McDonald, R. L.Derivatives Markets. Boston: Addison Wesley, 2003.

    McDonald, R.L. and D.R. Siegel. Investment and the Valuation of Firms When There is an Option to Shut Down,International Economic Review, Vol. 26 (1985), pp. 331-349.

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    McDonald, R. L. and D. R. Siegel. The Value of Waiting to Invest, Quarterly Journal of Economics, Vol. 100(1986), pp. 707-727.

    Mello, A. S., J. E. Parsons and A. J. Triantis, An Integrated Model of Multinational Flexibility and FinancialHedging,Journal of International Economics, Vol. 39, No. 1/2 (1995), 27-52.

    Mun, J.,Real Options Analysis: Tools and Techniques for Valuing Strategic Investments and Decisions. John Wiley& Sons, Inc. (2002).

    Myers, S.C. Determinants of Corporate Borrowing,Journal of Financial Economics, (1977), pp. 147-175.

    Myers, S.C. Finance Theory and Financial Strategy, Midland Corporate Finance Journal(Spring 1987), pp. 6-13.

    Myers, S.C and C.D. Howe, A Life-Cycle Financial Model of Pharmaceutical R&D, Program on thePharmaceutical Industry Manuscript, Massachusetts Institute of Technology (April 1997).

    Schwartz, Eduardo S., Patents and R&D as Real Options, Anderson Graduate School of Management at UCLAFinance Working Paper # 12-01 (2001).

    Schwartz, E.S. and L. Trigeorgis (editors),Real Options and Investment Under Uncertainty: Classical Readings and

    Recent Contributions, Cambridge: MIT Press, 2001.

    Schwartz, E.S. and C. Zozaya-Gorostiza, Valuation of Investments in Information Technology as Real Options,Anderson Graduate School of Management at UCLA Finance Working Paper (2000).

    Sender, Gary L. Option Analysis at Merck,Harvard Business Review (January/February 1994), p. 92.

    Siegel, D. R., J. L. Smith, and J. L. Paddock. Valuing Offshore Oil Properties with Option Pricing Models,Midland Corporate Finance Journal(Spring 1987), pp. 22-30.

    Smit, H., Analysis of Infrastructure Investment: The Case of European Airport Expansion, Erasmus UniversityWorking Paper, 1998.

    Smith, J.E. and K.F. McCardle, "Valuing Oil Properties: Integrating Option Pricing and Decision AnalysisApproaches." Operations Research 46 (1998), 198-217.

    Smith, J.E. and R. Nau, Valuing Risky Projects: Option Pricing Theory and Decision Analysis, ManagementScience, Vol. 41 (April) (1995), 795-816.

    Smith, K. W. and A. J. Triantis, The Value of Options in Strategic Acquisitions, in Real Options in CapitalInvestment, ed. L. Trigeorgis, Westport, Ct.: Praeger (1995), 135-149.

    Stonier, J. and A.J. Triantis, Natural and Contractual Real Options: The Case of Aircraft Delivery Options, in RealOptions Applications, ed. A. Micalizzi and L. Trigeorgis, (Milan: Bocconi University Press), 1999, 159-195.

    Titman, S., Urban Land Prices Under Uncertainty,American Economic Review, Vol. 75 (1985), 505-514.

    Triantis, A.J.,Real Options Database,www.rhsmith.umd.edu/finance/atriantis, 2003.

    Triantis, A.J. and A. Borison, Real Options: State of the Practice, Journal of Applied Corporate Finance, Vol. 14(2), Summer 2001, 8-24.

    Triantis, A.J. and J. E. Hodder. Valuing Flexibility as a Complex Option, Journal of Finance, Vol. 45 (1990), pp.549-566.

    http://www.rhsmith.umd.edu/finance/atriantishttp://www.rhsmith.umd.edu/finance/atriantishttp://www.rhsmith.umd.edu/finance/atriantishttp://www.rhsmith.umd.edu/finance/atriantis
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    Trigeorgis, L.,Real OptionsManaging Flexibility and Strategy in Resource Allocation. Cambridge, Mass.: MITPress, 1996.

    Trigeorgis, L. and S. Mason. Valuing Managerial Flexibility,Midland Corporate Finance Journal(Spring 1987),pp. 14-21.

    Williams, J.T. Real Estate Development as an Option,Jou


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