A DYNAMIC CAPITAL BUDGETING MODEL OF A PORTFOLIO OF RISKY MULTI-STAGE PROJECTS

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1 A DYNAMIC CAPITAL BUDGETING MODEL OF A PORTFOLIO OF RISKY MULTI-STAGE PROJECTS Janne Gustafsson, Tommi Gustafsson, and Paula Jantunen Abstract This paper presents a linear programming model in which a portfolio of projects is modelled as decision trees. It is also shown how scenario thinking can be combined with a capital budgeting model in order to permit uncertainty considerations. Furthermore, the programming of an option to wait and synergies of two projects is described. Key words: Capital budgeting, decision analysis, portfolio optimisation, linear programming, project selection, resource allocation. 1 INTRODUCTION During the past few years, there has been a growing interest towards valuation of research and development (R&D) projects. For instance, R&D investments are regarded as an option to market introduction in the real options theory, and the research in this field is extensive. Several authors including Amram & Kulatilaka (1998), Luehrman (1998), Dixit & Pindyck (1994), and Perdue et al (1999) have illustrated the applications of real options in this context. However, Faulkner (1996) points out that the options thinking leads basically to the same approach as decision trees (see French, 1986). While most of the present literature is focused on the valuation of a single R&D project, the interactions between individual projects and the constitution of the overall portfolio are left unconsidered. However, the interactions can account for a large part of the eventual value and risk of the portfolio. Morris et al (1991), for example, illustrate how riskier projects can be better than less risky ones, since their expected values are typically higher and investors can decrease risk by diversifying their portfolios. Moreover, a negative correlation between outcomes of different projects leads to the reduction of the overall variance and risk. 1

2 The usual approaches such as decision trees and real options do not determine how decisions concerning a portfolio that involves multi-stage projects should be made. Neither do conventional capital budgeting models (cf. Luenberger, 1998, and Martino, 1995) that often regard investment decisions as now or never choices and that seldom take uncertainty into account. Still, risky portfolio decisions are encountered, e.g., in the selection of pharmaceutical R&D projects where the need for appropriate selection frameworks is great (cf. Sharpe & Keelin, 1998). There are five issues regarding the portfolio selection and project interactions that are not addressed entirely by the current theories (cf. Martino, 1995): 1) The variance and the risk of the overall portfolio decreases as the number of projects increases. 2) Outcomes of projects are often correlated. The variance and the risk of the overall portfolio depends strongly on the correlations of the selected projects. 3) Projects may attain synergies among each other. By selecting similar projects, the value of each increases (by raising market potential or by reducing development time and costs). 4) There can be several ways to carry out the project. For example, investing the maximum amount of funds may not yield the best cost-benefit ratio (cf. Sharpe & Keelin, 1998). Moreover, sometimes delaying the project a month or two decreases risk and increases value. This is equivalent to the exercising of an option to wait in the real options literature. 5) There may be some non-profitable activities that improve the portfolio by either reducing risk or increasing the value of projects. The portfolio selection issues described above are also interrelated. For example, since achievement of synergies requires usually focusing the projects on the same field, the correlation between the projects increases as well. This, in turn, results in increase of variance and risk of the overall portfolio. Moreover, a small company, such as a typical biopharmaceutical R&D firm, may not be able to start a large number of projects due to scarcity of good projects and limits of budget and technical know-how, which also increases the portfolio s variance. Three possibly conflicting goals can be identified in the management of a project portfolio. First, the portfolio has to maximise the expected total net present value. 2

3 Second, the variance of the NPV of the portfolio needs to be minimised. Finally, the inherent uncertainty about the future has to be acknowledged such that no decision that unnecessarily ties our hands is made. This gives us more flexibility to act should something unexpected come up in the future. Flexibility is also discussed in conjunction with real options (see Amram & Kulatilaka, 1998, and Dixit & Pindyck, 1994). First and second goals are natural requirements from the financial theory, and they form also the basis of the conventional Markowitz portfolio theory (Luenberger, 1998). The third one is the precautionary principle (Stirling et al, 1999) of risk management and it is a typical optimality condition whenever sequential contingent decisions need to be made. In the literature, decision trees have proven quite successful in the modelling of single projects (French, 1986, and Clemen, 1996). Therefore they are also a natural project model in portfolio optimisation, although their formulation as decision variables and linear constraints of an optimisation program is not straightforward. However, while decision nodes are quite easy to model with decision variables, chance nodes pose substantial problems, since enumeration of the combinations of all outcomes of every chance nodes into different scenarios makes the program overwhelmingly large. There are basically two possible approaches to the modelling of chance nodes: 1) expected value and 2) scenarios. Conventional decision trees use dynamic programming and expected value to determine the value of the decision tree (French, 1986). The expected value of a chance node is obtained by multiplying the outcomes of the branches with their probability. However, while expected values do fine with single projects, their utilisation in a portfolio model may result in bias, since outcomes are only reduced by a multiplier instead of regarding the different realisations as separate events that also affect the decisions in other projects. Moreover, the expected cash flow stream is not necessarily any of the possible cash flow streams, but instead it is an average that is obtained if the chance node had resolved infinitely many times. In R&D each uncertainty is often unique and the use of expected values may lead to insufficient precaution for small revenues or an unnecessary large cash reserve when the revenues are higher than expected. Scenarios are often a better way of modelling chance nodes than expected values. For example, Lahdelma & Iivanainen (1996) and Korhonen (2000) have utilised 3

4 dynamic scenario optimisation in their financial linear programming models. Bunn & Salo (1995) have also considered the use and the construction of scenarios but within a different context. Scenarios are formed from the combinations of the outcomes of the uncertainties. However, the total number of scenarios is the product of the number of outcomes of the individual uncertainties. This leads to an exponential increase in the number of scenarios (and consequently in the number of decision variables) as the number of uncertainties increases. The number of chance nodes defining the scenarios has therefore to be restricted to a suitable amount, which is the most severe limitation confronted with the use of scenarios. Nonetheless, if the project portfolio is subject to only one or two major uncertainties, the combinations of their outcomes can be modelled with scenarios and other, minor uncertainties with expected values. The uncertainties that are modeled by scenarios are called scenario uncertainties in this paper and those dealt with expected value are referred to as project uncertainties, since they do not affect the decisions of other projects. The rest of this paper is structured as follows. Section 2 presents the theoretical background of the portfolio model. In Section 3 the basics of the portfolio model are presented. Then, the practical application of the model is considered in Section 4. In Section 5 some remarks regarding the model are discussed, and a software implementation of the model is presented in Section 6. The paper is summarised in Section 7. 2 THEORETICAL BACKGROUND The portfolio model has its roots in decision theory, financial theory, scenario analysis, and linear programming. While decision trees are used in the modelling of projects, the financial theory provides us with natural decision criteria for the evaluation of the project portfolio. Finally, linear programming is combined with scenario analysis to constitute the foundation of the portfolio model. 2.1 Decision Analysis In the literature, multi-stage projects are usually modelled with decision trees (cf. Clemen, 1996, French, 1986, Faulkner, 1996, and Sharpe and Keelin, 1998). They 4

5 are a conventional way of handling uncertainty, and as described in the following sections they can be applied in linear programming model with a few restrictions. 2.2 Net Present Value Net present value (NPV) is the fundamental tool for evaluating and comparing investment opportunities (Luehrman, 1997, Luenberger, 1998, and Brealey & Myers, 1996). NPV is based on two principles: A dollar today is worth more than a dollar tomorrow and A safe dollar is worth more than a risky one. Net present value is calculated as follows: NPV = T t t= 0 CF ( t) i= 1 (1 + r i ) (2.1) where CF(t) is the estimated cash flow in the year t, and r i is the discount rate for the year t. The discount rate can be chosen to correspond to the risk-free interest rate, a risk adjusted interest rate, or the opportunity cost of capital. The basic NPV rule is to invest in projects with a positive NPV, and prefer the project with the highest NPV (Brealey and Myers, 1996). When cash flows are uncertain, expected net present value is typically utilised (French, 1986). The variance of net present value can be used as a measure of risk. 2.3 Linear Programming Linear optimisation techniques have been widely used both in industry and academic institutes since the introduction of the simplex method in the 1950 s. Standard linear programming can be extended into mixed integer programming (MIP) models that permit the use of integral variables. In finance, linear programming is used typically in capital budgeting models. It enables the solving of the optimal decisions for these problems. However, uncertainty is seldom incorporated in capital budgeting models. 5

6 3 PORTFOLIO MODEL The portfolio model is presented here in three steps. First, the basic aspects of the portfolio model are introduced. Second, the decision tree approach to the modelling of multi-stage projects is presented. Finally, the objective function and the constraints forming the optimisation model are described. 3.1 Principles of the Portfolio Model The project portfolio model presented in this section is a linear capital budgeting model, which is based on an incremental formulation of a decision tree. In comparison with standard capital budgeting models (see Luenberger, 1998), the portfolio model differs in six aspects: 1) The time dimension is modeled explicitly. Therefore cash flow streams can be utilised as such and their net present value need not be calculated beforehand. 2) The projects are modelled with decision trees and therefore they may incorporate sequential decisions and uncertainties. 3) Scenarios can be used to assess decisions under different outcomes of the scenario uncertainties. 4) Any number of resources in addition to cash is permitted. The extension from the basic model is straightforward. 5) The projects can be allowed to be chosen more than once (cf. multiple decisions). 6) Prerequisites and restrictions for individual projects can be set by simple linear constraints. Many capital budgeting models also enable these restrictions. The formulation of some central restrictions is discussed in Section 4. The portfolio model is constructed in four steps. First, the period length and the time horizon of the model are defined. Second, the resources needed in the model are determined as are their weight in the objective function. Third, the uncertainties forming the scenarios are identified and the scenarios are constructed. Finally, the projects, their decision nodes, uncertainties, interactions, and cash flows under different scenarios are elicited. On the basis of this information the decision tree of 6

7 each project under each scenario is constructed. The next sections take a closer look at the portfolio model Time Horizon The model is divided into T+1 separate time periods whose cash inflows and outflows are calculated on the basis of the decisions made in the projects. Each decision in a project produces a stream of cash flows beginning from the period it is made up to the time horizon. The model does not rule out decisions that produce cash flows to the past, but such decisions are counterintuitive. Each period, the available cash has to suffice, i.e. the sum of the initial cash at the beginning of the period and the cash flows of that period has to be greater than zero Resources In addition to money, other resources, such as man-years, can be defined. They extend the model in three ways. First, each decision in a project produces a resource flow stream (like a cash flow stream) of each defined resource type. For example, the investments may produce streams of both cash flows and negative man-years. Second, the objective function becomes a weighted sum of the NPVs of the resource flow streams (usually the weight of non-monetary resources is set to zero). Finally, the constraints that ascertain the sufficiency of the resources are written with respect to each defined resource. Five properties characterise each resource: 1) Weight in the objective function. 2) Discount rate or equivalently the term structure of interest rates by which the NPV in the objective function is computed. 3) Transfer rate the proportion of surplus resources transferred to the next period. For example, the transfer rate of excess man-years is zero whereas it is for cash 1 + deposit interest rate. 4) Initial amount at each period. The initial amount of monetary resources is usually positive only at the beginning of the first period and zero in the 7

8 following periods. In contrast, the amount of available man-years is acquired at the beginning of each year. 5) Unit of measure Scenarios The resolution of scenario uncertainties divides the time axis into alternative futures as presented in Figure 1. The occurrence of each event is not determined by the time only but instead by both the time and the scenario in which the event happens. When scenarios are applied, events take place in time-scenario-space, which can be described as a tree of possible time lines. Every time period of the model is divided into scenarios that exist at that time, and separate sets of constraints and decision variables are defined for each scenario. For example, before the first scenario uncertainty resolves, there exists only one scenario, the base scenario. After the first scenario uncertainty resolves there are as many parallel scenarios as there are outcomes in the first scenario uncertainty. q 1-q Scenario 1.1 t = t 2..T p Scenario 1 t = t 1..t 2 Scenario 1.2 t = t 2..T Base Scenario t = 0..t 1 1-p r 1-r Scenario 2.1 t = t 2..T Scenario 2 t = t 1..t 2 Scenario 2.2 t = t 2..T t 1... T-1 T Figure 1. A scenario tree. 8 Time t t 2

9 Similarly, when the second scenario uncertainty resolves, the number of parallel scenarios is given by the product of the number of outcomes of the scenario uncertainties. Since scenarios branch the time axis, the projects decision trees are also separated in a similar way as a chance node branches the conventional decision tree (see Figure 1). Therefore scenarios in the portfolio model can be regarded as a multiproject extension of the chance node of the conventional decision tree. When scenarios are applied, the objective function is extended into a probability weighted sum of the NPVs of all the possible scenario paths (the way from the beginning to the time horizon through some realisations of scenario uncertainties), which is equivalent to the taking of expected value over the scenario uncertainties. 3.2 Model of a Single Project The model of a project is similar to a conventional decision tree. However, each branch of every node produces a set of resource flow streams instead of an instantaneous cash or resource flow like it is commonplace with standard decision trees (cf. French, 1986, and Clemen, 1996). Linearity of the model, however, restricts the decision trees into a quite simple, incremental format Tree Node Types Four different kinds of nodes can be distinguished: start, decision, chance, and end node. The start node automatically starts the project and produces resource flow streams without a prior decision. Decision and chance nodes are similar to those in conventional decision trees. The branches of a decision node are divided into two categories: into go branches that are associated with a decision variable and into a drop branch that is selected if no other branch is selected (decision variables of all the go branches are zero). A drop branch terminates the decision tree. Each decision node has exactly one drop branch but it can have several go branches. Finally, an end node terminates the tree in the same way as it does in normal decision trees. 9

10 Go Go Go z 1 z 2 z 3 z n Drop Drop Drop... Go Drop Figure 2. A decision tree with n consecutive decisions Decision Tree in Linear Programming Since chance nodes can be reduced away by taking expected values, we first derive the project decision tree model for a sequence of binary decision nodes. Let us assume that we have a sequence of n decision nodes as illustrated in Figure 2 and that there is a binary decision variable z i (in increasing order in time) associated with each decision node. Each decision node has two alternatives, either to continue to the next decision node (go branch, z i = 1) or to select the drop branch (z i = 0). Each branch ends up with an arbitrary cash flow stream. A linear model is of the form f z r ) = a + a z + a z + a z ( a n z n (3.1) where a i :s are freely chosen parameters and z i :s are decision variables. To assure consistency we must have z child z parent, (3.2) where z child refers to the decision variable associated with a child decision node of another, parent decision node. Parent decision node is, in turn, associated with the decision variable z parent. The inequation (3.2) ascertains that a subsequent node cannot be selected if the previous one has not been chosen. The inequation (3.2) also explains why a drop branch terminates the tree: it has no decision variable of its own to be referred to in the subsequent nodes. Furthermore, the restriction (3.2) limits the number of possible combinations of z i :s to n+1. Let us consider only one time period for which the decision tree produces cash flows. We realise that there are exactly n+1 free parameters in the linear model and just as many cash flows that are defined in the sequential binary decision tree. If we added a binary decision node to any of the ending branches, we would have a model 10

11 with n+2 free parameters and n+2 defined cash flows. Similarly, we can add any number of binary decision nodes to the tree without setting the number of free parameters and that of arbitrarily defined cash flows off balance. Therefore, for any kind of a decision tree with only binary decision nodes a linear function that assumes exactly the defined cash flows can be found. In the previous example we usually wish to branch the decision tree by adding additional nodes to some of the drop branches. However, since drop branches terminate the tree this would seem impossible. Fortunately, there is no need to restrict us to binary decision nodes with only one go branch. If a decision node has more than two branches, each additional branch must have a decision variable of its own. Let us denote the jth branch (let the drop branch be the number 0) of the decision node i with j z i. Regardless of the number of branches in a decision node, there is a constraint that restricts the total number of selections to the upper limit of the node: m j=1 z j i U i (3.3) where U i is the upper limit of decisions in the decision node i. While a tree of decision nodes can be constructed easily, a chance node does not contain any decision variables and therefore it cannot branch the tree without violating linearity. However, we can use a chance node as an affine transformation that takes an expected value. Such a chance node has only one go branch and one or more drop branches. The go branch scales the values of subsequent nodes by its probability and then the total probability weighted value of all the drop nodes is added to result. Nevertheless, as mentioned above, this may result in bias in the rest of the portfolio, since expected value may be far from the eventual outcome. Furthermore, the scenarios enable the branching of the decision tree. Therefore A z Figure 3. A decision tree a single decision. B 11

12 $ D $ F z 1 $ B p 1-p z 2 $ C $ E $ A Figure 4. A decision tree with two decision nodes and a chance node. scenarios should be used whenever they do not enlarge the model too much Derivation of the Parameters of the Linear Model Let us consider a decision tree with a single go/drop-decision, which is illustrated in Figure 3. The variable representing the decision is denoted by z, the cash flow from a positive decision by A and the cash flow from a negative decision by B. Clearly, the cash flow CF from the decision tree can be formulated as CF = A z + B (1 z). (3.4) Next we present an illustrative example in which we show how the resulting cash flow stream of a simple sequential decision tree with two binary decision nodes at both ends and a binary chance node in between is calculated as a function of decision variables z 1 and z 2. Figure 4 illustrates the example. Let A, B, C, D, E, and F be arbitrary cash flow stream vectors that spread from the period 1 to the time horizon T. Assume that the probability of the go branch in the chance node is p. Then, the cash flow stream of the decision tree produces as a function of binary decision variables z 1 and z 2 is CF ( z + z 1, z2 ) = A + ( A + B + (1 p) C + p( D + E)) z1 + p( E F) 2 We can check that if z 1 = z 2 = 0, then CF = A. If z 1 =1 and z 2 = 0, then CF = B + (1- p)c + p(d + E), which is the right result. Finally, if z 1 = z 2 = 1, then CF = B + (1- p)c + p(d + F), which also results in the right value. The derivation of the function f is based on incrementality. The parameter values are calculated from the beginning of the decision tree assuming that the values of subsequent decision variables are zero. For example, if all the decision variables are 12

13 Cash flow A A A A Go Go Go... z 1 z 2 z 3 z n Drop Drop Drop Go Cash flow B B B B Drop Figure 5. A decision tree with n identical go/drop-decision nodes. zero, then CF should be equal to A. Therefore the constant vector term is A. Then, we proceed to the next decision variable and adjust the parameter value such that it increases or decreases the value of the function just to the right amount. A simple computer procedure that calculates the parameters of the linear function defining the decision tree can be devised rather straightforwardly Multiple Decisions The model can be extended to handle decisions on amounts by altering the decision variables upper bound. Let us consider a decision tree in which n identical, successive go/drop-decisions so that both go and drop branches continue to the next decisions (illustrated in Figure 5). The cash flow from a positive decision is A and the cash flow from a negative decision is B. The number of positive decisions is denoted by z, and its upper bound by U, which is equal to n (0 z n). The total cash flow from the decision tree, CF, can be expressed as CF = A z + B (U z). (3.5) Equation is similar to Equation (3.4), and effectively n consecutive decisions are reduced into a single decision on the amount of positive decisions. In this way multiple decisions can be modelled with a single decision variable Continuous Decisions A decision variable need not be an integer variable, as assumed this far, but it can be also continuous as standard decision variables in linear programming. If so, the cash flow from the decision is a linear combination of the cash flows of the go and drop branches (instead of being one of them as with 0/1 variables). 13

14 3.3 Portfolio Model Objective Function The basic objective function of the portfolio model is the probability weighted sum (over scenarios) of the weighted sum (over resources) of discounted resource flow streams. In a simple case of one scenario and resource (cash) the objective function is the net present value of the cash flow stream the overall portfolio produces. The overall cash flow at time t is calculated with the formula r CF ( z, t) = J j= 1 r CF ( z, t), (3.6) j where J is the number of projects and CF j (z, t) is the cash flow of the project j in the period t when the decisions z are made. Cash flows are linear functions of decision variables as discussed in the previous section, and therefore the overall cash flow is also linear. The net present value of the cash flow stream defined by (3.6) is given by r NPV ( z) = T t t= 0 i= 1 r CF( z, t), (3.7) (1 + r d ( i)) where r d (t) is the discount factor (e.g. a short rate) in the period t. Since discounting only scales the cash flows, the NPV remains linear. Next we take other resources into account. Let NPV r be the net present value of the resource r calculated with the formulae (3.6) and (3.7) and w r the weight of the resource r in the objective function. The weighted sum of the net present values is then given by R r r RNPV ( z) = w NPV ( z), (3.8) r= 1 r r where RPNV is the weighted sum of resource net present values and R is the number of defined resources. Finally, the objective function is obtained by taking a probability weighted sum over all the possible scenario paths. Let S be the number of possible scenario paths, 14

15 RPNV s the weighted sum of resource net present values under scenario path s, and p s the probability of the scenario path s. Then the objective function s i given by as the expected RPNV S r r ERNPV ( z) = p RNPV ( z). (3.9) s= 1 s s By combining formulae (3.6)-(3.9) the objective function can be expressed as r max ERNPV ( z) = S R T J s= 1 r= 1 t= 1 j= 1 w p CF r s r j, r, s ( z, t), (3.10) where CF j,r,s (z, t) is the flow of the resource r of the project j in the period t under the scenario path s when the decisions z are made Constraints The primary constraints of the model assure the sufficiency of the resources each period. In addition to these, there are also several constraints from the projects decision trees as described in Section 3.2. Let C r (t) be the initial amount of the resource r in the period t, α the transfer rate multiplier from the previous period, CF r,s (z, t) the overall flow of the resource r under the scenario path s in the period t calculated similarly to (3.6). Then A r r r z, t) = α ( CF ( z, t 1) + A ( z, t 1)) C ( ) (3.11) r, s ( r, s r, s + r t is the amount of the resource r available at the beginning of the period t (A r,s (z, 0) = CF r,s (z, 0) = 0) under the scenario path s. By using (3.11) the primary model constraints can be expressed as CF r r z, t) + A ( z, t) 0 r s t. (3.12) r, s ( r, s The constraints (3.12) can be determined explicitly by beginning from the period 1 and computing the values of A r,s by using the recursive formula (3.11). 15

16 4 APPLICATION OF THE MODEL In this section we describe how the portfolio model can be applied in the modelling of synergies, deferring, and project prerequisites. We denote projects with p i, where i is the index of the project. The jth decision of the ith project is denoted by z i,j, which is binary unless otherwise noted. The term event is used to refer to any node of a decision tree describing a project. Events happen at some fixed time instant. Although this far (for the sake of simplicity) we have assumed that the portfolio consists of projects, it can actually include any kind of financial assets, e.g. bonds, that can be modeled with decision trees and resource flows. However, in order to avoid unnecessary confusion we continue to refer to the assets of the portfolio as projects. 4.1 Deferring projects It is not always the best choice to start a project when it first becomes possible, e.g., due to budget limitations. The question is when one should start the project. In the real options literature the decision to defer a project is given much attention, and it is referred to as an option to wait (Dixit & Pindyck, 1994). Let s consider two projects, p 1 and p 2, which are identical in every aspect except that every resource flow and event of p 2 happens one period later than in p 1. Effectively, p 2 is the same project as p 1 expect that it is deferred by one period. The starting decisions of the projects are z 1,1 and z 2,1. By constraining z 1,1 + z 2,1 1 (4.1) only one of the projects can be selected. By including the constraint (4.1) in the portfolio with the two projects, we have an option to defer the project p 1 by one period. If we select z 1,1 = 1, we start the project at once, and with z 2,1 = 1, we choose to defer it by one period. With both decision variables at 0, we do not start the project at all. If more deferring options are required, for example, an option to wait two periods, one may create more deferred duplicates of the project p 1, and include their starting decisions in the constraint (4.1). In this way, all deferring decision can be modelled. 16

17 In the previous example, we assumed that deferring the project would not affect the project s cash flows. This may not be true in reality, since, for example, an early entry to the market is often desirable. Therefore, the deferred projects p 2, p 3, etc., may have to be adjusted to reflect the effects of later entry. These changes can include adjustments to resource flows, adjustments to decisions, new decisions, adjustments to probabilities in chance nodes, and entirely new uncertainties. 4.2 Synergies and Interactions Some projects may have synergies with (or negative effects on) each other. For instance, projects that produce complementary deliverables most likely result in improvement in sales. On the other hand, if the products are partial or complete substitutes, the market success of one may cannibalise the sales of the other (Martino, 1995). Synergies can be formulated in a similar way as the deferring of a project. Let us consider two projects, which have synergies. If one chooses to start only one of them, the selected project is either p 1 or p 2 (with no synergies). If both of them are started the projects p 1 * and p 2 * are selected instead of p 1 and p 2. The projects p 1 * and p 2 * are modifications of p 1 and p 2 that include the effects of synergies. We denote the starting decisions of the four projects by z 1,1, z 2,1, z 1*,1, and z 2*,1. The situation can be described with the following linear constraints: z 1,1 + z 1*,1 1, (4.2a) z 2,1 + z 2*,1 1, z 1*,1 = z 2*,1. (4.2b) (4.2c) If there are more than two interdependent projects, the above constraints can be extended straightforwardly. For each project, a constraint similar to (4.2a) and (4.2b) is added. If synergies apply only when all the projects are chosen, all decision variables with an asterisk (starts a project with synergies) must be equal to each other; this is equivalent to the constraint (4.2c). If synergies depend on the mix of projects selected, the situation is more complicated, and one has to create a separate projects which reflect the effects of synergies in different cases. This may quickly lead to a very large number of projects representing different synergies. 17

18 4.3 Prerequisites In some cases projects cannot be started unless some other project has been started (or more typically the second cannot start before the first has finished). For example, an applied research project that exploits results of a basic research project cannot be started if the basic research project was not started in the first place. Similar technological interactions are common in R&D project selection (Martino, 1995), and their implementation is considered in this section. We use the term prerequisite to refer to the dependence of a decision on another decision. We denote the dependent decision by z d and the decision on which z d depends by z i. If z i has to be 1 (e.g. the project is started) in order to allow z d to assume the value 1, the situation can be modelled by the following linear constraint: z d z i (4.3) If z i should be 0 (e.g. the project is not started) that z d could be 1, we use the following constraint: z d 1 z i. (4.4) If z d depends on more than one project, say, z 1, z 2, z 3, z n, (n decisions) the constraint (4.3) can be extended as follows: n z d z 1 + z z n (4.5) Similarly, if all the decisions z 1,, z n must be 0, that z d could be 1, the constraint (4.4) can be extended to n z d n z 1 z 2 z n. (4.6) If the upper bounds of z-variables are not limited to 1, the constraints (4.5) and (4.6) have to be modified. First, n is replaced by the sum of the upper bounds of z 1,, z n. If z d itself has an upper bound greater than one, one must consider how many times (once, twice, as many times as its upper bound states, ) z d can be chosen when the prerequisites are met. If the number is greater than 1, one should consider how the selecting a subset of z 1,, z n affects the number, and then redefine the formulas (4.5) and (4.6). Redefining the constraint (4.6) poses some additional problems, but they are not addressed in this paper. 18

19 Many interactions can be modeled by using prerequisites, but the interactions that depend on the outcome of uncertainties must be modelled by using scenarios. 5 REMARKS 5.1 Estimation of Cash Flows and Uncertainties Estimation of cash flows and uncertainties is often a demanding challenge, and poor accuracy of these parameters may vitiate the benefits of the portfolio model. With regard to this consideration and the general purpose of the model, some situations when the application of this model is appropriate can be identified: There is a need to do trade-offs between projects. This is typically a result of budget restrictions and a large variety of projects available. The cash flows and the uncertainties can be estimated reasonably well. 5.2 Chance Nodes Since chance nodes are modelled by taking expected values, the result is not usually equal to any of the outcomes, which may complicate risk management. However, the problem can be avoided by modelling uncertainties with scenarios. 5.3 Minimisation of Variance Minimisation of variance with respect to scenario uncertainties cannot be achieved with a linear model. Moreover, the formulation of the variance of NPV of the portfolio as a function of decision variables is a complex and challenging task. 5.4 Precautionary Principle The precautionary principle of risk management (Stirling et al, 1999) can be taken into account with the use of scenarios. By identifying critical uncertainties and constructing scenarios on the basis of them, the decisions prior to these uncertainties are made such that they maximise the expected net present value over all the possible scenarios. 19

20 6 SOFTWARE IMPLEMENTATION The prototype software Capital Budgeteer which implements the presented model is discussed briefly in this section. Capital Budgeteer provides a user interface for collecting data, creates and solves appropriate linear programs, and returns the results to the user. Some illustrative windows are shown in Figures 6 and 7. Capital Budgeteer makes it possible to test and use the model present in this paper in real life situations. We chose to build a stand-alone software since standard spreadsheet applications proved insufficient and too inflexible for the calculation of parameters (especially those of the linear function representing the decision tree) and for the formulation of scenarios and primary constraints. The graphical user interface of Capital Budgeteer makes it easy to form the project portfolio models the user need not know how the model is actually transformed into a linear optimisation model. The software is able to both solve the model and calculate the probability distribution of the outcomes, which enables value at risk (VaR) (see Schachter, 1997) and critical probability considerations. It offers presently many possibilities for graphical evaluation (see Figure 7), and we plan to implement sensitivity analysis in the future. The models are easily saved, modified, and solved. Still, the software is at prototype stage and it requires several enhancements to the user interface before it may be used more widely. Figure 6. The properties of the portfolio as implemented in Capital Budgeteer. 20

21 Figure 7. One of the result windows of Capital Budgeteer. 7 CONCLUSIONS There are three goals in the selection of a project portfolio: 1) maximising of expected net present value, 2) minimising of variance of net present value, and 3) the precautionary principle. We presented a linear portfolio model that maximises the expected net present value of the portfolio and which is in alignment with the first and the third goal. Due to linearity of the model variance with respect to scenarios uncertainties (second goal) cannot be taken into account. There are three primary differences in the presented model in comparison with standard capital budgeting models. First, the explicit time axis enables scheduling of projects, and assigning cash flow estimates accurately for discounting. Second, the projects are modelled with decision trees that permit consecutive decisions and uncertainty considerations. Finally, the use of scenarios in handling uncertainties allows decision trees to continue differently in each of the scenarios and improves possibilities to take uncertainty into account. 21

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23 Perdue, R. K., McAllister, W. J., King, P. V., Berkey, B. G. (1999). Valuation of R and D Projects Using Options Pricing and Decision Analysis Models, Interfaces, Vol 29, Iss 6, November-December, pp Schachter, B. (1997). A Guide to Value at Risk, Financial Engineering News, Vol 1, Iss 1, August. Sharpe, P., Keelin, T. (1998). How SmithKline Beecham Makes Better Resource Allocation Decisions, Harvard Business Review, March-April, pp Stirling, A., Renn, O., Klinke, A., Rip, A., Salo, A. (1999). On science and precaution in the management of technological risk, EC Forward Studies Unit, Final Report. 23