INCORPORATING RISK IN A DECISION SUPPORT SYSTEM FOR PROJECT ANALYSIS AND EVALUATION a

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1 INCORPORATING RISK IN A DECISION SUPPORT SYSTEM FOR PROJECT ANALYSIS AND EVALUATION a Pedro C. Godinho, Faculty of Economics of the University of Coimbra and INESC João Paulo Costa, Faculty of Economics of the University of Coimbra and INESC ABSTRACT This paper addresses the incorporation of risk in a Decision Support System (DSS) for project analysis and evaluation, focusing on the use of real options and decision trees. We briefly present these methods and discuss their incorporation in a DSS. Next, we argue that the use of criteria other than the financial value may often make sense. We discuss the importance of using time as a criterion, and we present an approach that allows the simultaneous use of time and financial value in classical and real option decision trees. Finally, we present an illustration example of this new approach. I. INTRODUCTION We, along with some other members of our Faculty, are developing a group decision support system (GDSS) for project analysis and evaluation the AGAP (Aid to Groups of Analysis and evaluation of Projects) system - that is partially designed and developed. The system incorporates the most common financial methods for project evaluation, including equivalent worth measures, rate of return measures, profitability indexes, payback measures and accounting measures. It is also prepared for the use of multicriteria methods for project evaluation, including multi-attribute utility (MAU), ELECTRE and PROMETHEE methods. We are now including some techniques and methods for the incorporation of risk in project value. This paper will focus on the incorporation of real options and decision trees in the system. First, we will examine decision trees and real option models, and we will define some guidelines for their incorporation in a DSS. Section II presents these methods and discusses which type of support should be provided in order to facilitate their use within a DSS for project analysis and evaluation. Real options and decision trees will usually consider projects from a financial perspective. While this perspective is considered the most important by capital budgeting textbooks, other perspectives, and other criteria, may also be important. Project management textbooks, for example, stress that time and consumption of limited resources may be very important. Also, there may be certain factors that cannot be incorporated in the financial value of the project and are very important to decide whether or not the project should be undertaken. Consider, for example, a highly profitable project that may cause a great deal of pollution. This project may damage the company image and its long-term prospects in ways that cannot be easily foreseen or quantified. So, it makes sense for the company managers to consider pollution as an independent evaluation criterion for this project. Time is one criterion that will often be very important. In a construction project, for example, there will usually be a deadline. If the company does not meet the deadline, it may have to pay some compensations, and its image may be damaged in a way that is hard to quantify. So, managers may often want to consider time as an independent criterion. It makes thus sense to support the use of multiple criteria in project evaluation and selection. Section III presents a new approach that uses multiple criteria in classical decision trees and real option decision trees (decision trees that are evaluated using the binomial model for option valuation). It particularly focus on the use of two criteria - time and financial value - but we think it can be easily extended to other criteria. This new approach allows the decision-maker to identify all the non-dominated alternatives 1, thus letting him/her use any multicriteria method to choose among them. Section IV presents an illustration example of this approach. In this section we consider a situation in which time and financial value are the relevant criteria to evaluate a particular project. Then we use our approach to identify the non-dominated alternatives. Finally, we present our conclusions in section V. II. DECISION TREES, REAL OPTIONS AND THEIR INCORPORATION IN A DSS This section briefly presents decision trees and real options, and discusses their incorporation in a DSS.

2 Decision trees provide a way of representing sequences of decisions and uncertain events through time, so that decisions made today take proper account of what can be done in the future (see Magee, 1964 and Brealey and Myers, 1991). Decision trees are usually evaluated from their leaves to the root nodes, using a process called rolling back. This process consists on working out which are the best decisions at the latest stage, and then choosing the best decisions at previous stages, supposing that later decisions will be optimal. The possible outcomes from uncertain events are characterised through probabilities, and the value of an event node will be the expected value of its descendants the sum of its descendant values weighted by their probabilities. A tree structure must be defined in order to correctly handle decision trees in a DSS. The correct definition of a decision tree may be a tedious and difficult process. Thus, it will be convenient to incorporate in the DSS some tools to support it. Often, the branches are equal (or very similar) for all the nodes at the same level, or all the branches of the same kind lead to similar nodes. When this happens, even if the tree grows very large (see the example in the chapter 3 of Hertz and Thomas, 1984), it can be defined using a limited number of rules. These rules may be of the type: All branches at level 5 that correspond to high demand lead to a decision node with two branches. Thus, the DSS should somehow support the use of such rules in the definition of decision trees. One possible way to implement this feature is allowing that, after the definition of a node and its corresponding branches, the node may be copied to all the branches that follow a certain rule, which is defined using a given syntax. The usual copy and paste tools may also be very important in the definition of decision trees. In a next stage, we will try to allow the interactive definition of decision trees through questions like Do all branches at the present level lead to identical nodes?, Is the node an event node or a decision node? or What is the cash flow associated with this branch?. Many authors acknowledge that decision trees treat project risk incorrectly (see, for example, Trigeorgis and Mason, 1987 and Trigeorgis, 1996). In the presence of different options, the project risk will change, and so should the discount rate change. However, classical decision trees consider the same discount rate independently of the choices being faced. In the presence of choices or options, managers actions will limit the downside potential and/or increase the upside potential of investments, thus the claims will no longer be symmetric. In this situation, it will be very difficult to assess a correct discount rate. Furthermore, as Trigeorgis (1996) points out, sometimes the correct discount rate will be very unstable, presenting additional problems. To overcome these problems, it has been proposed the use of Option Pricing Theory to evaluate projects that incorporate options thinking of projects as real options. The use of option valuation requires the existence of an underlying asset. The option value will be contingent on the value of that asset. Options can be categorised into call options and put options. Call options give the right to buy the underlying asset for a pre-specified price - the exercise price - during a specified period; put options give the right to sell the underlying asset for a pre-specified exercise price during a specified period. Option valuation relies on two equivalent techniques. The first is the replication of the option by a portfolio consisting on the underlying asset and risk-free bonds, and using an arbitrage argument to calculate the option value. The second is the risk-neutral valuation: assuming that investors are risk-neutral, accordingly adjusting the underlying asset growth rate and using the risk-free interest rate to discount the claims on the adjusted underlying asset values. When events have a limited number of outcomes, as is the case with decision trees, the binomial model for option valuation is used. The value of the options represented by this model may be calculated using risk-neutral valuation. First, the event probabilities are adjusted to become risk-neutral probabilities, or the probabilities that would occur in a world where investors were risk-neutral. Then, expected cash flows (calculated according to the risk-neutral probabilities) are discounted at the risk-free interest rate. We will now show an example of the use of the binomial model for real option valuation. Consider that, to explore a copper mine, a company must make an immediate investment of $5m. The company estimates that, in the next year, there is a 40% probability of a rise in copper prices and a 60% probability of a decline in copper prices. If copper prices rise, the project will be worth $8m in one year, otherwise it will be worth $4m. The investment cannot be deferred but it can be abandoned in one year for a salvage value of $5m this abandonment option can be seen as a put option with an exercise price of $5m and expiration in one year. The risk-free interest rate is 5% and similar publicly traded copper mines present a 1-year expected rate of return of 25%. Obviously the project should be abandoned if copper prices decline and continued otherwise. If we were to use classical decision trees, the project NPV would be NPV = -$5m+(0.4*$8m+0.6*$5m)/1.25 = -$40000, so the project wouldn t be undertaken. If we were to use option valuation, we would start by calculating the value of the copper mine without the abandonment option: PV = (0.4*$8m+0.6*$4m)/1.25 = $4.48m. The risk-neutral

3 probability of a rise in copper prices would then be p =($4.48m*1.05-$4m)/($8m-$4m) = 17.6%, and the present value of the project would be NPV = -$5m+(0.176*$8m+( )*$5m)/1.05 = $ So, in spite of what classical decision trees prescribed, the project should be undertaken, because its true NPV is positive. Using the binomial model to calculate option values is basically a process of resolving decision trees. The main difference between them is that the binomial model also requires the price formation process of the underlying asset. In the DSS, we will use the same infrastructure for classical decision trees and for the binomial model, so that the user must only define which of them should be used at the evaluation phase note that the binomial model may only be used if the price formation process of the underlying asset has been previously defined. In order to simplify the definition of the decision tree, we will allow the semi-automatic definition of some common types of options (abandonment option and deferment option, for example), by inputting some information. The price formation process of the underlying asset will be defined in an independent tree. This tree will usually have only one node and two branches, meaning that the process will be the same in all moments. However, we will allow this tree to have more levels, corresponding to different processes in different moments or situations. The Black-Scholes model is a continuous time extension of the binomial model. This model assumes that the underlying asset value will no longer have a limited number of values, but it will follow a continuous generalised Wiener process (with no value jumps ) and its value on the option expiration date will be log-normally distributed. The inputs for the model are the current value and the variance in the value of the underlying asset, the exercise price and the time to expiration of the option, and the risk-free interest rate. Sometimes there will be a depreciation in the underlying asset value (for example for lost cash flows, in the case of real options, or lost dividends, in the case of financial options) that can also be included as an input for the model. The implementation of the Black-Scholes model in a DSS will be straightforward, since the Black-Scholes formula may be directly applied to the inputs in order to calculate the option value. However, some support will be provided to the definition of the inputs, since this is usually a difficult task. Particularly, we will allow the use of historical data to estimate the variance of projects in specific sectors. Also, some tools will be provided to effectively estimate the depreciation factor that should be used when there are lost cash flows or when there is an opportunity cost for deferring the project. Sometimes, the user may want to use the binomial model as an approximation to continuous-time evaluation of real options. This may happen when the Black-Scholes model cannot be used for example in American put options (see Trigeorgis, 1996) or when the underlying asset value doesn t follow a log-normal distribution. Notice that, in this situation, the possibility of interactive semi-automatic definition of a real option decision tree will be even more important because, in order to achieve a good approximation, small time increments must be used, and this will lead to very large trees. In this situation it will also be important to provide further support to the definition of the price formation process of the underlying asset. We will allow the automatic definition of the underlying asset value tree from the characteristics of its price formation process; particularly, we intend to include some pre-defined common processes. For further reference on real options, see Trigeorgis (1995, 1996), Trigeorgis and Mason (1987), Dixit and Pindyck (1995), Luehrman (1998) and Brealey and Myers (1991). This section presented some methods for the incorporation of risk in project value: decision trees and real options. It also discussed the incorporation of these methods in a DSS. We focused on the type of support that should be provided, and also discussed some implementation issues. In the next section we will present a new approach to the use of multiple criteria in classical and real option decision trees. III. USING MULTIPLE CRITERIA IN CLASSICAL AND REAL OPTION DECISION TREES Capital budgeting textbooks often emphasize the use of financial value as the most important, or even the only criterion that should be considered in project evaluation. On the other hand, project management textbooks mention other important criteria, like time or consumption of limited resources. Also, other factors may sometimes be important to assess whether or not a project should be undertaken. The environmental consequences of the project or the quality of the products resulting from the project, for example, may influence the company image and therefore management may want to consider these factors in project selection. Furthermore, it may be very difficult, or even impossible, to quantify the direct and indirect influence of these

4 factors in the cash flows. So, it is important that a DSS for project analysis and evaluation allows the users to consider criteria other than the resulting from the use of financial methods. Time will often be an important factor in project selection. A construction project, for example, must usually meet a specified end date. If that deadline isn t met, the company may face a demand for monetary compensations and its image may be damaged. Although monetary compensations may often be easily included in the cash flows, damages to the company image are harder to quantify. There may also be some benefits from an early conclusion of the project, like the possibility of undertaking other projects or seizing other opportunities. Since these benefits and damages may be very hard to quantify and even to foresee, managers may want to consider time to completion resulting from a certain strategy as an independent criterion. Competitive interaction will many times provide other important reasons to use time as a criterion. A company will often have an option to defer a project, in order to gather more information. Project deferment will sometimes be the optimal strategy in the absence of competition but, if other companies may undertake similar projects, competitive issues may influence the decision about the investment timing. If the company knows that competitors already possess the necessary technology, or if it can predict when the competitors will possess the necessary technology to undertake similar projects, the company can use game theory and, assuming that competitors will follow an optimal strategy, it can define its own best strategy and the best timing for the investment. Unfortunately, companies don t often know which technology their competitors already possess and cannot predict when they will have the technology to undertake a particular type of project. So, in order to deter competitive entries, to gain a competitive advantage or to avoid losses resulting from an early competitive entry, companies will want to undertake a project as soon as possible, while trying to maximise its financial value companies will thus want to use time and financial value in the definition of their strategies. Other important reasons for the use of time as an independent criterion are related to the personal preferences of the decision-makers. Sometimes an early conclusion of the project may increase the prestige and power of the project manager, who will thus have an incentive for speeding up the project. Other times the management salary package will include some options on company stock. If the managers believe that an early conclusion of some projects (even at the expense of additional costs) may lead to a rise in company shares, they may try to have the projects concluded before the exercise date of their options. Such factors may motivate the use of time as an independent criterion in the analysis and evaluation of investment projects. The direct relation between financial value and time makes the simultaneous use of both criteria particularly hard. Difficulties arise from the growth of uncertainty in profits or costs along with time. Consider the following example about a project to build a plant: if managers are unsure about whether the plant will take 2 or 3 years to be built, the project cash flows will differ according to the time it will take to build that plant. If the plant takes 3 years to be built, not only the first operational cash flow will occur later, but it will also be subject to one more year of uncertainty in prices. We propose a new approach to deal with this kind of problems using classical and real option decision trees. We will focus on the use of two criteria: time and financial value. However, we think that this approach can be easily used with other criteria. We will assume, in the remainder of this section, that our criteria are financial value, in the form of net present value (NPV), and time; we ll also assume that we re maximising financial value and minimising time. First, we will propose a new type of node to simplify the definition of multicriteria decision trees that include time as a criterion. This node will represent the passage of time and the corresponding changes in financial value. We will call it the time passage node. n Figure 1: The time passage node. The node in figure 1 will represent the passage of n units of time. The branches following this node (we ll refer to them as output branches) will represent each possible state after the passage of that time. Assuming that the relevant criteria are time and financial value, and that financial value follows a binomial model, we ll have n+1 output branches. The same is true if financial value depends on an underlying variable (a commodity price,

5 for example) whose value follows a binomial process. In this situation, supposing n=2 and using variable T to represent time and variable V to represent financial value, the node would be equivalent to the sequence of event nodes shown in figure 2. V V*u 2 2 V V*u 2 V V*u*d V V*u*d and T T+2 for all branches V V*d 2 V V*d 2 Figure 2: The equivalence between a time passage node and a sequence of event nodes. In figure 2, we consider that the financial value (V) may either became V*u or V*d when a time unit passes. In order to keep the figure simple, we didn t include the probabilities of the output branches. Should the initial probabilities be p u and p d (p u +p d =1), the probabilities of the output branches would be (from the upper to the lower branch): p u 2, 2*p u *p d and p d 2. Notice that the use of the time passage node provides a much more compact representation of the passage of time than the use of uncertainty nodes. The evaluation of the decision tree will be a two-step process. In the first step, time increments and cash flows (or other value increments) are forwarded to the leaves, in order to calculate the criteria values for each leaf. In the second step, event probabilities are adjusted (if necessary) and the tree is rolled back. Since we re using multiple criteria, the rolling back process will differ from the usually used. Hertz and Thomas (1983) present an overview of some methodologies for the evaluation of multicriteria decision trees, based on multi-attribute utility theory (MAUT). These methodologies include the reduction of leaf values to equivalent outcomes that differ only in terms of the values of one criterion (based on Raiffa, 1968) and the use of additive and multiplicative MAU models (based on Edwards, 1976 and Keeney and Raiffa, 1976). Some important drawbacks of the former methodology are the potential necessity of defining equivalent outcomes for all the leaves (except the one being used as a reference) and the possibility of the decision-maker not being able to make the required trade-off assessments 2. Also, all these methodologies are based on MAUT models, not allowing the use of other multicriteria methods that do not assume the existence of an utility function. We will try to define an approach that can be used not only with MAUT-based methodologies but also with other multicriteria methods. Our approach focus on the identification of the non-dominated alternatives 3, in order to allow the decision-maker to choose one of these alternatives using any multicriteria method. Thus, the process of rolling back the tree allows the identification of all the non-dominated alternatives, and not only one best alternative. First we will consider the aggregation of criteria values across event nodes. Since financial value and time may follow different aggregation rules, adjusted probabilities for these criteria may differ. We may even end up with three different probabilities in the same branch: the basic (initial) probability, a value-adjusted probability and a time-adjusted probability. The financial value that corresponds to a given event node will always be the sum of the values corresponding to its branches, weighted by their value-adjusted probabilities, regardless of whether classical or real option evaluation are being used. Only the adjustment of probabilities will depend on the kind of evaluation being used. If the classical evaluation of decision trees is being used, then there is no need to adjust probabilities (value-adjusted probabilities will be equal to the initial probabilities). If the binomial model for real option valuation is being used, probabilities must be adjusted to risk-neutral value-adjusted probabilities according to the underlying asset pricing process. When the binomial model is being used, a new problem will arise. Some event nodes may not correspond to events that influence the value of the underlying asset. For example, uncertainty about whether a factory will take 18 or 20 months to be built will not influence the value of a similar publicly traded factory. So, the question arises: when the binomial model is used, how shall financial value be aggregated across nodes that don t

6 influence the value of the underlying asset? Such events that do not influence the underlying asset value will probably be unique to the project, and shall thus be treated as unsystematic risk. This means that the probabilities associated with such nodes shall not be adjusted 4 for the calculation of financial value, even if the binomial model is being used. In most situations, only the event nodes that correspond to the passage of time will require the adjustment of probabilities, since the value of a publicly traded underlying asset will only change if time passes. Time may be aggregated across event nodes in several different ways, depending on the situation that the decision-maker is facing. Sometimes, average time to completion or average delay relative to a specified completion time will be enough; in this situation, time-adjusted probabilities will be equal to the initial probabilities. In other situations, only the worst possible case the maximum time - will be of interest. This may happen, for example, when the decision-maker is only interested in alternatives that guarantee project completion in a specified time. In this situation, time-adjusted probabilities will be 100% for the branch with the longest time (or for one of the branches with the longest time) and 0% for the other branches. We think that the decision-maker does not usually want to use none of the previous approaches we think that an uncertain time will, in most situations, be considered equivalent to a fixed time between the average and the maximum of the uncertain times. For example, if there is a 50% chance of project completion in 12 months and a 50% chance of project completion in 24 months, the decision-maker will probably consider this time distribution equivalent to a fixed time between 18 (the average) and 24 (the maximum) months. So, we propose another, more general, approach to aggregate time across event nodes. This approach is similar to the use of the binomial model for option valuation, and it relies on the use of certainty equivalents. We start by asking the decision-maker for a certainty equivalent of a specified uncertain time. Suppose that the situation shown in figure 3 is found on the tree being evaluated. Assuming linearity, we can calculate certainty equivalents for all event nodes with the same probabilities. If the decision-maker provides a certainty equivalent of 2.5, then he is implicitly adjusting probabilities to (50%, 50%). So, each time the pair of probabilities (60%, 40%) comes up, probabilities will be adjusted to (50%, 50%). p - =60% T=2 p + =40% T=3 Figure 3: Example of a binomial node with uncertain time. p - T=m T=M Figure 4: A generic binomial node with uncertain time. p + We will now consider the generic process shown in figure 4. Let m be the shortest time, M the longest time, p - and p + their corresponding probabilities, and CE the certainty equivalent time provided by the decision-maker. The adjusted probabilities will be p - =(M-CE)/(M-m) and p + =(CE-m)/(M-m). Assuming linearity, we can replace every instance of (p -, p + ) by the time-adjusted probabilities (p -, p + ). Notice that the choice of the longer time as the certainty equivalent leads to the maximum time to completion approach, and the choice of the average time as the certainty equivalent leads to the average time to completion approach. So, we can say that maximum time to completion and average time to completion are particular cases of this approach. According to this approach, the decision-maker must provide a certainty equivalent for each combination of probabilities (p -, p + ) in the tree. So, if there are many different combinations of probabilities in the tree, the decision-maker will be asked to provide many certainty equivalents. This may be seen as a serious drawback for this approach. However, we don t think this may be an important drawback. First, because most trees will only

7 have few combinations of probabilities if the value of the underlying asset follows a constant binomial process, all nodes corresponding to uncertainty about its price will have the same probabilities. Moreover, if there are many different probabilities in the tree, some numerical methods (like interpolation methods) may be used to estimate certainty equivalents from the ones provided for nodes with different but similar probabilities. A last problem may be raised by the use of time: how should time be handled when some situations don t lead to project completion? Usually time to completion should be infinite in these situations. However, if the probability of time being infinite is not zero, the aggregation of an infinite time will lead to an infinite time. This might lead to some odd situations. Suppose we have two different alternatives with the same financial value. One of these has a 95% probability of project completion in one year and a 5% probability of non completion. The other has a 80% probability of project completion in 2 years and a 20% probability of non completion. Should time be aggregated as infinite, the two alternatives would be presented as equivalent to the decision-maker, although the first is clearly better than the second. Some different approaches may be used to avoid such pitfalls. One possible approach consists on the use of two different criteria to handle time when there is the possibility of not completing the project: the probability of project completion and time to completion when the project is completed. Another such approach is the use of a criterion that aggregates both the probability of project completion and time to completion: this type of aggregation may be easily achieved by using a finite value to represent an infinite time 5. We will now consider the evaluation of decision nodes. As we previously explained, we want the tree evaluation process to provide all the non-dominated alternatives. This means that all the decisions involving non-dominated alternatives must be made at the root node. We will use three different rules to accomplish this. The first rule is: two consecutive decision nodes will be merged. This means that, when there are consecutive decision nodes, we will consider one choice among all the alternatives represented by these nodes, and not consecutive choices among some of those alternatives. Figure 5 shows the use of this rule. A A B B C C Figure 5: Example of the use of rule 1. A, B and C are different alternatives. The use of this rule allows us to directly choose among alternatives A, B and C, not being forced to choose between A and B before we consider alternative C. The second rule is: eliminate all dominated alternatives in a decision node. This rule will be valid under the assumption that the aggregation of criteria across event nodes will never lead to criteria values above the largest value nor below the smallest value being aggregated. So, we can eliminate one alternative in a decision node if there is another one with a larger or equal financial value and a shorter or equal time, one of the inequalities being strict. Should, in figure 5, alternative B be dominated by alternative C (should it have a smaller or equal financial value and a longer or equal time, one of the inequalities being strict), then it would be eliminated from the decision node. The third rule is: if there is an event node before a decision node, then postpone the decision by considering all possible combinations of decisions. Figure 6 shows the use of this rule.

8 Notice that, if the lower branch would have a decision node with two alternatives, the total number of alternatives in the resulting tree would be four. This rule may cause a large growth in the number of alternatives. So, it is important to use the second rule in each decision node, in order to prevent the number of alternatives from becoming too large. C A A C B B Figure 6: Example of the use of rule 3. A, B and C are different alternatives. C There is no need to explicitly modify the tree in order to apply these rules. In the example shown in figure 5 we will only have to consider that there is a choice among alternatives A, B and C. This way it will be possible to eliminate alternative C before making a choice between alternatives A and B, if the former alternative is dominated by one of the latter ones; it will also be possible to eliminate alternative B (or A) before choosing between A and B, if that alternative is dominated by alternative C. These rules allow us to delay all decisions involving non-dominated nodes until all event nodes are evaluated. Then, any multicriteria method may be used to choose among these non-dominated alternatives. This section discussed the use of multiple criteria in classical and real option decision trees. We presented a new approach that identifies all the non-dominated alternatives, allowing the decision-maker to use any multicriteria method to choose among them. IV. AN ILLUSTRATION EXAMPLE This section provides an illustration example for the use of multiple criteria in decision trees. We will assume that a company intends to build a new plant. This new plant will allow the company to introduce a new type of product, which no one is yet producing. The managers expect that some other companies will, in the next couple of years, try to market similar products. If competition arises after the product is marketed, managers expect the competitors to have a hard time and probably give up in the short term. However, if a similar product is introduced before the company starts production, things may become very tough for the company. So, the managers want to build this new plant before competition arises but they have no idea when that will happen. The managers only know that they want the plant built as fast as possible. Thus, they think that the time until starting production and the financial value (as measured by the NPV) will be the relevant criteria. The decision to immediately start construction has already been made and two different processes, A and B, may be used to build the plant. Construction will take two years under process A and one year under process B. Process A requires a $ investment and process B requires a $ investment. Managers expect the plant to generate perpetual cash flows. According to some preliminary studies, if the plant were already built, it would have generated a $ cash flow in the present year. It is also expected that each year there will be a 50% chance of a 20% rise and an equal chance of a 20% decline in the value of the cash flows. We will assume that the discount rate for projects with similar risk is r=10% and the risk-free interest rate is r F =6%. We will further suppose that, after the plant is built and before starting production, management has the option to make some changes in the plant. These changes will take one more year. If these changes are made, the plant will generate a certain annual cash flow of $ Since an option is involved, the binomial model will be used to evaluate the project NPV. The decision tree for this project and the corresponding results are shown in figure 7. All monetary values are in thousands of dollars, and we will also hereafter consider all monetary values in thousands of dollars. After building the tree, we started by calculating the criteria values corresponding to each leaf. We will now explain how the criteria values were calculated for leaf L1 (a similar process was used for the other leaves). In this leaf, process A is used to build the plant and no modifications are made after the plant is built. Under

9 process A the plant will take 2 years to be built, so the time corresponding to this leaf will be T=2. Since the value of the cash flows rises twice for L1, the expected cash flow for each operating year will be CF=1000*1.2 2 =1440. This will lead to a plant value in year 2 of PV 2 =1440/10%= Since we are using the binomial model, we will use the risk-free rate to discount the plant value in year 2, thus obtaining a NPV= / =5816. Figure 7: The complete decision tree of the example. Values are in thousands of dollars and times are in years. After calculating the criteria values for all the leaves, we eliminated the leaves corresponding to dominated alternatives. Leaves L2, L4 and L8 were eliminated since they were dominated by L1, L3 and L7, respectively. The decisions between L5 and L6 and between L9 and L10 cannot be made at this moment, since none of these leaves correspond to dominated alternatives. So, these decisions must be delayed. In order to aggregate alternatives across the time passage nodes, we must assess the time- and value-adjusted probabilities of the branches, represented in the figure by pv and pt, respectively. We will start with the value-adjusted probabilities. Suppose that the expected cash flow for a given year is CF and the project value in that year is PV=CF/0.1. If the cash flow rises to 1.2*CF, the project will be worth PV 1 + =1.2*CF+1.2*CF/0.1=1.32*PV one year later. If the cash flow declines to 0.8*CF, the project will be worth PV 1 - =0.8*CF+0.8*CF/0.1=0.88*PV. According to the binomial model the adjusted probability for a rise will be pv + =(PV*(1+r F )- PV 1 - )/(PV 1 + -PV 1 - )=( )/( )=41%, and the adjusted probability for a decline will be pv - =1-pV + =59%. In order to assess the time-adjusted probabilities, managers would be asked to provide a certainty equivalent for an uncertain time. Suppose that, faced with a 50% probability of T=1 and a 50% probability of T=0, they

10 provide a certainty equivalent of EC=0.6. Then the adjusted probability for the longest time will be pt + =(EC- 0)/(1-0)=0.6 and the adjusted probability for the shortest time will be pt - =1- pt + =0.4. Once the time- and value-adjusted probabilities are assessed, we can aggregate the values of non-dominated alternatives across the time passage nodes. By using the third rule of the previous section, we will have two alternatives in the upper tree branch and another two alternatives in the lower branch. The alternatives in the upper branch correspond to the combinations of leaves (L1,L3,L5) and (L1,L3,L6), and the alternatives in the lower branch correspond to the combinations (L7,L9) and (L7,L10). The combination (L1,L3,L5) leads to a longer time and a smaller NPV than the combination (L7,L10), so we can eliminate the former alternative. This would lead to the final set of three non-dominated alternatives. Any multicriteria method could now be used to choose one of the resulting non-dominated alternatives. This method must be chosen according to the needs of the managers, in order to aggregate their preferences. This section presented a simple example of the use of our approach to the use of multiple criteria in classical and real option decision trees. We have shown how our approach can be used to generate all the non-dominated alternatives when time and financial value are the relevant criteria. V. CONCLUSIONS This paper addressed the incorporation of risk in a DSS for project analysis and evaluation, focusing on the use of real options and decision trees. We started by presenting decision trees and real options, and we discussed their incorporation in a DSS for project analysis and evaluation. Next we argued that the decision-maker may want to consider not only the financial value but also other criteria, because some important factors may be difficult, or even impossible to include in the financial value of a project. We explained that time will often be a particularly important criterion, for many reasons. These reasons include potential benefits arising from an early conclusion of the project, losses arising from a late conclusion, competitive issues and personal preferences of the decision-makers. These issues will often be very important to the decision-makers and also impossible to quantify in the financial value of a project. So, in such situations, time and financial value should be used as independent criteria. We defined a new approach that allows decision-makers to use time and financial value in classical or real option decision trees. This approach identifies all the non-dominated alternatives, allowing the use of any multicriteria method to choose among them. This way, the decision-maker can analyse the different non-dominated combinations of criteria values before deciding how they will be aggregated. This analysis can provide the decision-maker better insights into the possible project behaviour, and may also help him/her select the best fitted multicriteria method to choose among the alternatives. Finally, we also presented an example of our approach. We defined a project for which time and financial value are the relevant criteria, and we used our approach to identify all the non-dominated alternatives. ( a ) The authors wish to thank Dr. Jane Binner and an anonymous referee for their helpful comments. This research was supported by PRAXIS/PCSH/C/CEG/28/96.

11 ENDNOTES 1. An alternative is non-dominated if none of the other alternatives is better or equal on all the criteria and better on at least one criterion. 2. Hertz and Thomas (1993) refer that Serious questions about the ability of the decision-maker to make the required trade-off assessments have been raised in the literature. 3. An alternative is non-dominated if none of the other alternatives is better or equal on all the criteria and better on at least one criterion. Assuming the use of time and financial value, an alternative will be non-dominated if none of the other alternatives has a shorter or equal time and a larger or equal financial value, with a strict inequality for at least one of the criteria 4. Since the risk associated to such nodes will not be relevant to the investors, risk-neutral value-adjusted probabilities will be the same as the initial probabilities. 5. To see how this can be done, let s assume that we are working with times in the range of 1 to 3 years, that we ll use 100 years to represent an infinite time and that we re using the weighted average to aggregate across event nodes. The situation where there is a 5% probability of non completion and a 95% probability of time being 1 year will have a certainty equivalent time of 5.95 (=0.05* *1) years, so it will be considered better than a 20% probability of non completion and 80% probability of time being 2 years (certainty equivalent time of 0.2* *2=21.6 years), and it will be considered worse than a certain time of 3 years. The number used to represent an infinite time should be fine tuned in order to correctly represent the importance that the decision-maker attaches to the probability of non completion.

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