A multistage stochastic programming approach for capital ...€¦ · A multistage stochastic programming approach for capital budgeting problems under uncertainty ... Risk is explicitly

IMA Journal of Management Mathematics (2012) Page 1 of 22doi:10.1093/imaman/dps018

A multistage stochastic programming approach for capital budgeting problemsunder uncertainty

Patrizia Beraldi∗, Antonio Violi, Francesco De Simone

Department of Electronics, Informatics and Systems, University of Calabria, 87036 Rende (CS), Italy∗Corresponding author: [email protected] [email protected] [email protected]

and

Massimo Costabile, Ivar Massabò, Emilio Russo

Department of Business Administration, University of Calabria, 87036 Rende (CS), [email protected] [email protected] [email protected]

[Received on 18 April 2011; accepted on 16 May 2012]10

This paper addresses the capital budgeting problem under uncertainty. In particular, we propose a multi-stage stochastic programming model aimed at selecting and managing a project portfolio. The dynamicuncertain evolution of each project value is modelled by a scenario tree over the planning horizon. Themodel allows the decision maker to revise decisions by decommitting from a given project if it shows anegative performance. Risk is explicitly assessed by defining a mean-risk objective function, where theconditional value at risk is used. A customized branch-and-bound method is also introduced for solvingthe proposed model. Extensive computational experiments have been carried out to validate the modeleffectiveness, also in comparison with other possible benchmark policies. The numerical results col-lected by solving randomly generated instances with the proposed branch-and-bound approach seems tobe encouraging.

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Keywords: risk management; stochastic programming; capital budgeting.

1. Introduction

Capital budgeting decisions are of primary concern in business since they may affect the company struc-ture. Managers are frequently asked to select a project portfolio meeting some specified organizationgoals while honouring constraints imposed by the management or external real-world restrictions.

The classical formulation of the capital budgeting problem was presented by Lorie & Savage (1955). 5

They proposed a multi-period model aimed at selecting a certain number of projects in a given basketwhich maximizes the total net present value subject to limited capital resources constraints. The basicformulation relies on the project selection, ignoring the possibility of gathering additional funds inthe market as well as of considering alternative investment opportunities. Later, in Weingartner (1963,1966a,b), the author extends the model by considering both theoretical and application issues. Many 10

other researchers (see Baumol & Quandt, 1965; Myers, 1972; Padberg & Wilczak, 1999) focus theirattention upon important related issues such as the choice of suitable objective functions, the selection ofappropriate discount rates, the inclusion of borrowing and lending decisions at different rates and so on.

All the models mentioned above are based on the assumption of perfect information concerningproblem parameters (e.g. future project outcomes), but such a hypothesis is meaningless in real-life

c© The authors 2012. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

2 of 22 P. BERALDI ET AL.

settings, because of the uncertainty affecting market conditions. As a consequence, deterministic cap-15

ital budgeting models have limited, if any, value as managerial tools for supporting financial decisionmakers.

With the aim of explicitly taking uncertainty into account, different formulations based on thestochastic programming framework have been proposed in the scientific literature. In particular, twomain approaches have been investigated. The first one relies on the use of chance constraints and it has20

in Byrne et al. (1968) and Sarper (1993) some of its main representatives. We also mention the recentcontribution of Beraldi et al. (2011) where uncertainty and risk are dealt by probabilistic constraintsjointly imposed on the limited budget restrictions and a mean-risk objective function. The approachbased on probabilistic constraints suggests a ‘proactive’ strategy in that the investment plan is definedin order to hedge, with a given confidence level, the potential losses due to future adverse events. The25

second approach is based on the recourse paradigm and defines a ‘reactive’ strategy aimed at correct-ing first-stage decisions on the basis of incoming information about the uncertain parameters. Amongothers, we mention Kira et al. (2000) where the authors propose a two-stage formulation where therecourse decisions are associated with shortage and surplus variables used to measure the violationof the stochastic budget constraints. The objective function maximizes the total expected net present30

value minus the expected recourse penalty cost. A multistage model based on a decision tree describ-ing the dynamics of individual projects was proposed in Gear & Lockett (1975). More recently, Meieret al. (2001) proposed a scenario-based dynamic optimization model for capital budgeting where thereal option valuation approach is used to model a project’s value. Even though both uncertainty anddynamic aspects of the capital budgeting problem are encompassed, the formulation cannot be framed35

as a multistage model since no specific recourse actions are considered. The main flexibility of such amodel stems from the possibility of deferring the decision to accept a certain project. In Costa & Paixao(2010), the authors extended the previous contribution by proposing a more computationally tractableformulation and a tailored heuristic procedure. Finally, we mention the contribution of Gustafsson &Salo (2005) where the authors apply the contingent portfolio programming framework to address the40

problem of selecting and managing a portfolio of research and development projects under uncertainty.The considered framework integrates decision analysis with portfolio optimization and allows the finan-cial planner to account for a wide range of risk measures.

It is worth mentioning that the capital budgeting problem shares some fundamental features with theAsset-Liability Management (ALM) problem (see, e.g. Consigli & Dempster, 1998; Mulvey & Ziemba,45

1998; Kouwenberg & Zenios, 2001 for an extensive review). Indeed, both projects and assets may beconsidered investment opportunities bearing a certain degree of risk due to the uncertain evolution ofthe market conditions. This evident analogy has led, in recent years, to the definition of mixed assetportfolios obtained by integrating traditional securities with research and development projects. Asshown in Fang et al. (2008) (see also the references therein), such a strategy may lower the risk of the50

overall portfolio and increase the potential for more benefits over the long term.In this paper, we propose a multistage stochastic programming model aimed at selecting and man-

aging a portfolio of projects. The uncertain evolution of project value is represented by a scenario treedefined in a discrete-time framework following the Amin (1991) approach. A distinctive feature of theproposed stochastic programming formulation is related to the possibility of taking stage-wise decisions:55

at each stage, recourse actions can be taken to modify the initial portfolio composition by deciding toterminate one or more projects because of negative evolution of market conditions. In addition, the pro-posed model explicitly assesses risk, which is measured by the Conditional Value at Risk (CVaR) on thepotential losses. This risk measure, as demonstrated by the results in Rockafellar & Uryasev (2000) andAhmed (2006), allows a more accurate measure of tail losses and admits a linear reformulation which60

MULTISTAGE FORMULATION FOR CAPITAL BUDGETING 3 of 22

makes the problem more computationally tractable. The proposed model can be suitably integrated intoa dynamic framework to allow an active project management. In fact, in current practice, beyond theproject investment decision, managers are called to periodically monitor the project portfolio evolution.If project performance is not aligned with business goals, or is below expectations, decision makersmay choose to decommit from a project and redirect available resources to other projects or investment 65

opportunities.The rest of the paper is organized as follows. Section 2 is devoted to the model development. After

describing the scenario generation approach used to model the uncertain evolution of the project value,we introduce the stochastic programming formulation. The proposed model belongs to the class of mul-tistage mixed-integer programming problems. It is widely recognized that the high modelling power 70

of the multistage paradigm comes at the expense of an increased computational complexity which pre-vents the use of general purpose software for solving cases of a large size. In Section 3, we introduce acustomized branch-and-bound method which exploits the specific problem structure. Preliminary com-putational results are reported in Section 4 where, in addition, we provide evidence of the model’s effec-tiveness in comparison with other possible benchmark policies. Some concluding remarks are provided 75

in Section 5.

2. Model development

Let us consider a planning horizon of T time periods and let t denote the corresponding time index.We denote by J the set of projects to evaluate and we assume (see Meier et al., 2001; Costa & Paixao,2010) that each project value evolves according to a discrete-time stochastic process. Thus, the entire 80

information structure can be represented by a scenario tree rooted at node 0. Each node n at time tencapsulates the state of the world uniquely characterized by information available up to time t. In ourcase, for each project j ∈ J , node n contains the project value Vjn and the corresponding cash flow Cjn

(by convention, inflows are considered positive whereas outflows are negative).To better clarify notation, in Fig. 1 we report a generic scenario tree. For each stage (or level) t, we 85

denote by St the corresponding set of nodes. With each node n is associated a unique parent denotedby an, and a state probability πn. Furthermore, we denote by P(n) the path connecting the root to noden. If n is a leaf node, then P(n) corresponds to a scenario and represents a joint realization of theuncertain parameters over all time steps t = 0, 1, . . . , T . For simplicity of notation, in what follows, weshall denote by S the set of leaves and by N the whole set of nodes. 90

2.1 Scenario generation

Scenario generation represents a critical issue in any stochastic programming formulation, since thequality of the generated input data affects the effectiveness and accuracy of the recommendations pro-vided by the model. Over recent decades, effort has been devoted to this topic and many scenario-generation techniques, often tailored for specific applications, have been proposed. Among others, we 95

mention historical approaches, Monte Carlo simulation techniques, forecasting and optimization-basedmethods (see, e.g. Kaut & Wallace, 2007 and the references therein).

In our application, we have chosen to deal with uncertainty affecting projects by representing theirevolution by means of a correlated scenario tree. Such an approach overcomes the limits of the tradi-tional discounted cash-flow techniques which rely on the strong assumption of certain project cash flow 100

(see Miller & Park, 2002).


Fig. 1. A scenario tree.

We propose a scenario generation procedure which relies on a binomial model derived through alog-transformed variation of the binomial option pricing schema reported in Amin (1991). Like the Tri-georgis binomial model (Trigeorgis, 1991), the considered approach has the nice feature of overcomingthe problem of negative probability for specific parameters, as may happen in Cox et al. (1979). This105

is a crucial aspect to look at since, typically, a relatively small number of time steps discretizing thecontinuous time process is considered, due to the high number of possible scenarios generated aggre-gating all the investment projects. The choice of Amin’s binomial approximation is further justified byits efficiency in pricing complex real options as in the dynamic optimization model we will proposein Section 3. Indeed, real investments often include more than one option simultaneously and the pric-110

ing of each option separately (as happens in McDonald & Siegel, 1985, 1986; Majd & Pindyck, 1987;Myers & Majd, 1990) would result in inappropriate evaluations since interactions among them are notconsidered.

In order to present the binomial model used to discretize the project dynamics over time, let usdenote by Vj, j ∈ J , the present value of the jth project. It may be seen as the market value of a claim on115

future cash flow if the firm were to get into the project at the present time.We suppose that the process describing the dynamics of the project value Vj has the form

dVjt

Vjt= rj dt + σj dzjt, (2.1)

where rj denotes the drift term and σj the instantaneous standard deviation,1 while zjt is a standardWiener process. The correlation among projects will be discussed later in this section.

1 The instantaneous standard deviation σj of the jth project may be estimated by one of the approaches presented in the literaturewhich is very extensive in this field. Just to name a few, we recall the approaches proposed by Smith (1995), Davis (1998), Kelly(1998), Cobb & Charnes (2004) and Godinho (2006), but a more comprehensive analysis of the proposed methodologies isprovided in Kodukula & Papudesu (2006).


It is straightforward to observe that the log-transformed process Xjt = log Vjt follows an arithmetic 120

Brownian motion in any time interval dt, i.e.

dXjt =(

rj − 1

2σ 2

j

)dt + σj dzjt. (2.2)

The continuous process is discretized by dividing the total project life into T equal subintervals oflength Δt. Then, at the end of each subinterval Δt, the corresponding discrete process X d

j varies by theamount

ΔX dj =

(rj − 1

2σ 2

j

)Δt + σjε

√Δt, (2.3)

where ε is a random variable with state space {±1}. In particular, we define a negative and a positive 125

variation as follows:

(ΔX dj )− =

(rj − 1

2σ 2

j

)Δt − σj

√Δt,

(ΔX dj )+ =

(rj − 1

2σ 2

j

)Δt + σj

√Δt

The mean and variance restrictions imply that the probability of each state equals 0.5 so that the firsttwo-order moments are matched and the discrete process is consistent with the continuous one. 130

The following scheme presents the computation of the jth project value for each node n of the tree.

For each level t = 0, . . . , T − 1For each node n ∈ St

Node generation: Create two childrendenoted by 2n + 1 and 2n + 2with probabilitiesπ2n+1 = πn

2 and π2n+2 = πn2 ;

Value generation : Set Vj 2n+1 = exp(

X dj n + (ΔX d

j )−)

and Vj 2n+2 = exp(

X dj n + (ΔX d

j )+).

Here Xj0 denotes the log transformation of the jth project value Vj0 at the root node.

2.1.1 The aggregated scenario tree. Starting from each project’s value dynamics, we may derive theaggregated scenario tree taking into account the possible correlations affecting the projects.

To present how to manage the possible project correlations, for the sake of clearness, we first limit 135

our analysis to the case of two correlated projects with dynamics V1t and V2t:

dV1t

V1t= r1 dt + σ1 dz1t and

dV2t

V2t= r2 dt + σ2 dz2t,

where z1t and z2t are Brownian motions showing a constant correlation ρ ∈ (−1, 1).


Since our aim is to develop a scenario tree that considers the joint evolution of the project val-ues, in general not evenly independent, we need to determine the joint project dynamics. The projectdependence is not easily managed in our framework because the joint transition probabilities may notbe obtained as the product of the marginal transition probabilities related to the values of the single140

projects. A desirable situation would be one where the project dynamics are independent of each other.With this aim, hereafter, we present a step-by-step procedure which allows us to obtain independent pro-cesses modelling project values, thus simplifying the construction of the aggregated scenario tree. Theprocedure may only be applied upon homoskedastic processes and this is the reason why, as detailed inthe following Step 1, we consider the logarithmic transformation of both processes V1t and V2t. Once145

the homoskedastic processes X1t = log V1t and X2t = log V2t have been obtained, the procedure is basedon the definition of a new Brownian motion, Pt, detailed in Step 2, and of an auxiliary process, Yt,reported in Step 3, where we also show that the stochastic part of Yt is driven by the Brownian motionPt. The auxiliary process Yt has the nice feature of being independent of the homoskedastic process X1t,thus simplifying matters for what concerns the calculation of the joint transition probabilities, which150

may now be obtained as the product of the marginal transition probabilities related to the values of thesingle projects. Step 4 details the discretization provided for the obtained independent projects, X1t andYt, while Step 5 shows how the discrete values assumed by the second homoskedastic process, X2t, maynow be obtained by combining the discrete outcomes of X1t and Yt. Finally, Step 6 details the construc-tion of the scenario tree which also illustrates the exponential inverse transform is also illustrated, which155

allow us to obtain the values of the original heteroskedastic processes, V1t and V2t, starting from one ofthe corresponding log-transformed processes, X1t and X2t.

The proposed procedure is summarized as follows.

Step 1. To obtain two homoskedastic processes starting from V1t and V2t, consider the log-transformedprocesses X1t = log V1t and X2t = log V2t with dynamics specified, respectively, by160

dX1t =(

r1 − 1

2σ 2

1

)dt + σ1 dz1t and dX2t =

(r2 − 1

2σ 2

2

)dt + σ2 dz2t.

Step 2. Define the process

Pt = 1√1 − ρ2

(z2t − ρz1t),

which is still Brownian motion. Indeed, Pt ∼ N(0, 1) since z1t, z2t ∼ N(0, 1) and, consequently,(z2t − ρz1t) ∼ N(0, 1 − ρ2). In addition, Pt has the nice feature of being independent of z1t, incontrast with z2t which has constant correlation ρ with z1t, i.e. Cov(z1t, z2t) = ρ. This is eas-ily proved: Cov(Pt, z1t) = E[Ptz1t] = E[(z1t/

√1 − ρ2)(z2t − ρz1t)] = (1/

√1 − ρ2)(E[z1tz2t] −165

ρE[z21t]) = (1/

√1 − ρ2)(Cov(z1t, z2t) − ρ) = 0.

It is also worth noting that the Pt dynamics are described by

dPt = 1√1 − ρ2

(dz2t − ρ dz1t).


Step 3. Consider now the auxiliary process

Yt = X2t − ρσ2

σ1X1t, (2.4)

for which dynamics are obtained by applying Ito’s formula in two-dimensions, resulting in

dYt = dX2t − ρσ2

σ1dX1t =

(r2 − 1

2σ 2

2

)dt + σ2 dz2t − ρ

σ2

σ1

[(r1 − 1

2σ 2

1

)dt + σ1 dz1t

]170

=[

r2 − 1

2σ 2

2 − ρσ2

σ1

(r1 − 1

2σ 2

1

)]dt + σ2(dz2t − ρ dz1t).

Consequently, the Yt dynamics results are captured by dYt = μy dt + σy dPt where

μy =[

r2 − 1

2σ 2

2 − ρσ2

σ1

(r1 − 1

2σ 2

1

)]and σy = σ2

√1 − ρ2.

Clearly, it is easy to prove that Yt is independent of X1t since Pt is independent of z1t.

Step 4. Discretize, through the binomial lattice presented above, the independent processes X1t and Yt,thus obtaining the discrete processes X d

1 and Y d . In particular, X d1 moves according to 175

ΔX d1 =

(r1 − 1

2σ 2

1

)Δt + σ1ε1

√Δt, (2.5)

and Y d moves according to

ΔY d = μyΔt + σyεy

√Δt, (2.6)

where ε1 and εy are random variables with state space {±1}.

Step 5. Discretize the process X2t through X d1 and Y d . The discretization is obtained by considering

that at the end of each subinterval of length Δt, both X d1 and Y d present two possible outcomes.

Consequently, because from (2.4) the continuous process X2t may be written as 180

X2t = Yt + ρσ2

σ1X1t,

at the end of each subinterval of length Δt, the corresponding discrete version X d2 increases or

decreases its value by one of the following amounts:

(ΔX d2 )++ = (r2 − 1

2σ 22 )Δt + σ2(

√1 − ρ2 + ρ)

√Δt;

(ΔX d2 )+− = (r2 − 1

2σ 22 )Δt + σ2(

√1 − ρ2 − ρ)

√Δt;

(ΔX d2 )−+ = (r2 − 1

2σ 22 )Δt + σ2(−

√1 − ρ2 + ρ)

√Δt; 185

(ΔX d2 )−− = (r2 − 1

2σ 22 )Δt + σ2(−

√1 − ρ2 − ρ)

√Δt.


Step 6. The scenario tree approximating the joint process (V1t, V2t) is obtained following the pseudo-code below, where in Value generation we use the inverse transform to obtain the values assumedby the original heteroskedastic processes, V1t and V2t, starting from one of the corresponding log-transformed processes, X1t and X2t.190

For each level t = 0, . . . , T − 1For each node n ∈ St

Node generation: Create four children denoted by4n + 1, 4n + 2, 4n + 3 and 4n + 4with occurrence probability, respectively,π4n+1 = πn

4 , π4n+2 = πn4 ,

π4n+3 = πn4 and π4n+4 = πn

4Value generation : Set the pair project values

for each child asV1 4n+1 = exp(X d

1 n + (ΔX d1 )−) and V2 4n+1 = exp(X d

2 n + (ΔX d2 )−−),

V1 4n+2 = exp(X d1 n + (ΔX d

1 )−) and V2 4n+2 = exp(X d2 n + (ΔX d

2 )−+),V1 4n+3 = exp(X d

1 n + (ΔX d1 )+) and V2 4n+3 = exp(X d

2 n + (ΔX d2 )+−),

V1 4n+4 = exp(X d1 n + (ΔX d

1 )+) and V2 4n+4 = exp(X d2 n + (ΔX d

2 )++).

Figure 2 illustrates the aggregate scenario tree in the case of two projects.It is worth mentioning that the procedure presented for two correlated projects may be extended

to the case of more than two projects correlated with each other. In this case, we have to manage avariance–covariance matrix and we have to resort to the Cholesky decomposition to manage the projectcorrelations contemporaneously. Furthermore, when the project correlation has value ρ = 1 or ρ = −1,195

the projects’ results will be, respectively, affected by the same source of risk (we will refer to suchcases as ‘similar’ projects) or affected by variables which have opposite evolutions, and, consequently,the scenario tree development is easier. In those cases, just two children are enough to describe thevalue generation for each node n. Following the considerations introduced above, we may considercorrelated groups of similar projects. In more detail, we may aggregate in each group the projects with200

an estimated high positive correlation (ρ ∼= 1) in the sense that for the same node of the scenario treewhen one project value grows the same happens for the others. Then, the aggregate scenario tree isdeveloped by considering the correlations among groups and managing them as explained above.

We finally observe that the cash flow associated with a project is strictly related to the project valueso that we define2 for each node n of the aggregated scenario tree the project inflow/outflow by the205

difference Cjn = Vjn − Gj, where Gj is a threshold value for the jth project exogenously determined.We observe that the procedure described above can generate very large scenario trees, especially

when the number of correlated groups of similar projects is high, thus leading to the definition of opti-mization problems that difficult to treat from a computational standpoint. To deal with such a difficulty,it is possible to resort to scenario reduction procedures (see, e.g. Heitsch & Römisch, 2009 and the210

references therein) which are aimed at reducing the size of the original scenario tree while trying topreserve all the relevant information.

2 Other definitions for the project cash flow may be considered in this framework without any restriction. An example is thedefinition for the project cash flow provided in Meier et al. (2001, p. 202) where it is managed as a perpetual option pay-off.

Antonio e Martina

Nota

that are difficult


Fig. 2. Scenario tree for two projects.

2.2 Mathematical formulation

The proposed stochastic programming formulation relies on the scenario tree defined in the previoussection. Given a limited budget Bt available at each time step t = 0, 1, . . . , T , the problem consists in 215

defining the optimal strategy to select and manage a portfolio of projects. With the aim of proposinga more general model to include in a dynamic rolling horizon setting, we assume the pre-existenceof the projects defined by the set I, in addition to the set of projects J . For each project k ∈ (I ∪ J),


the uncertain evolution of the project value (and the cash flow) is represented by the scenario tree. Wedenote by Fj the initial investment required by project j, j ∈ J , whereas at the initial time the pre-existing220

projects have a cash flow equal to Ci0, i ∈ I. For the sake of simplicity, we assume that each project hasa residual life longer than the planning horizon. The generalization to the case of shorter project lifeis straightforward, since in that case we have for each finishing project a terminal cash flow whicheventually includes the residual value at the final stage of the project life.

The stochastic nature which characterizes the capital budgeting problem has suggested the adop-225

tion of the multistage stochastic programming paradigm. With respect to traditional deterministicapproaches, multistage formulations allow us to define more flexible plans, since modifications of previ-ously defined decisions are possible if adverse conditions occur. More specifically, the decision processis discretized into a set of consecutive stages corresponding to the scenario tree time periods. In the firststage, which corresponds to the root node of the scenario tree, decisions regarding project selection are230

taken (here-and-now decisions). In later stages, corrective actions on the project portfolio compositioncan be defined according to new information concerning uncertain projects’ outcomes. These ‘recourse’actions are related to the possibility of terminating some projects if their performance can no longer beconsidered satisfactory. When a project is terminated, we assume that eventually a certain value couldbe recovered and we denote by Rj the recovery rate. Formally, the model variables, related to each235

project, are the following:

• xj: binary variable taking value 1 if project j ∈ J is selected and 0 otherwise;

• ykn: binary variable taking value 1 if project k ∈ (I ∪ J) is terminated at node n and 0 otherwise.

We consider both the possibility of investing the total positive cash flow generated by the selectedprojects and the net positive surplus between project cash flow and available budget in a risk-free asset,240

thus producing an additional amount of money eventually used in later periods. For each node n of thescenario tree, we denote by fn the corresponding decision variable and by γ the constant yield for a unittime period.

The model includes different types of constraints, accounting for both balance and logical condi-tions. A first set of constraints represents the financial balance between inflows/outflows and available245

resources:∑j∈J

Fjxj + f0 = B0 +∑i∈I

Ci0(1 − yi0) +∑i∈I

Ri0Fiyi0, (2.7)

fn = Bt +∑j∈J

Cjn

⎛⎝xj −

∑m∈P(n)

yjm

⎞⎠ +

∑i∈I

Cin

⎛⎝1 −

∑m∈P(n)

yim

⎞⎠

+∑

k∈(I∪J)

RkFkykn + fa(n)(1 + γ ) ∀n ∈ St, t = 1, . . . , T . (2.8)

Constraint (2.7) states that at the initial time, total outflow due to investments in new projects and in the250

risk-free asset must be matched by the available budget adjusted by the eventual (positive or negative)cash flow related to existing projects and by the eventual recovery due to projects terminated early.Equation (2.8) extends the balance for later stages, when there are no costs for investments in newprojects and cash flow for active projects are considered. The last term in (2.8) stands for the capitalizedvalue of cash investment in the ancestor node which we assume can be added to the available budget at255

the current node.


Another set of constraints is related to logical conditions among acceptance and decommitmentvariables:

∑m∈P(n)

yim � 1 ∀i ∈ I, ∀n ∈ S, (2.9)

∑m∈P(n)

yjm � xj ∀j ∈ J , ∀n ∈ S. (2.10) 260

Condition (2.9) imposes that an existing project can be terminated only once along each scenario,while constraint (2.10) states that an investable project can be terminated only if it has been previouslyaccepted.

Many other types of constraints may be added to the model, depending on the specific situation tobe managed. For example, the decision maker could be concerned about the relation among projects 265

identifying mutual exclusiveness conditions or about the presence of mandatory projects, or may wishto limit the number of active projects at each time stage.

The selection of the optimal strategic plan is driven by the choice of an appropriate objective func-tion that mathematically translates the decision maker goals, i.e. profit maximization and proper riskmanagement. Over recent decades, two main approaches have been adopted and developed by aca- 270

demics and researchers in order to take risk representation into account in decision processes underuncertainty: the stochastic dominance (see Levy, 1992) which defines theoretical axioms for risk aver-sion, and the mean risk (see Markovitz, 1987), which implements an intuitive trade-off analysis bymeans of two opposite criteria. As regards the first approach, a recent and very interesting stream ofresearch is the one related to the stochastic dominance constraints strategy (see Gollmer et al., 2008, 275

2011) for models with mixed-integer linear recourse, a class of formulations very effective in the finan-cial field (see Sriboonchitta et al., 2010).

On the other hand, as regards the mean-risk approach, several risk functionals have been proposed(see, e.g. Ahmed, 2006 and the references therein) starting from the variance that, as simple counterex-amples can show, might fail to discriminate between over-performance and under-performance, leading 280

to incorrect dominance relations (see Ruszczynski & Shapiro, 2006). In this paper, in order to assessrisk exposure, we have considered the CVaR on potential losses for a certain confidence level β (usually95%). This is a modern formulation which with respect to the well-known Value at Risk (VaR) allows amore accurate measure of tail losses. The overall objective function has the following structure:

max[(1 − λ)WT − λΦ], (2.11)

where Φ represents the risk measure and λ ∈ (0, 1) is user-defined parameter accounting for risk aver- 285

sion. The higher the value of λ, the more conservative, but also the less profitable, the decision-makerposition.

The first term of (2.11) represents the expected value of final portfolio wealth:

WT =∑n∈S

πnWn, (2.12)


which is defined starting from the portfolio wealth at each terminal node n of the scenario tree:

Wn =∑j∈J

Vjn

⎛⎝xj −

∑m∈P(n)

yjm

⎞⎠ +

∑i∈I

Vin

⎛⎝1 −

∑m∈P(n)

yim

⎞⎠ + fn. (2.13)

As described above, the risk is measured by the CVaR on potential losses. Even if the presence of integer290

variables makes the model discontinuous and non-convex, it is proven that stochastic integer programs,with the CVaR as risk measure, result in being computationally treatable (see Schultz & Tiedemann,2006).

For each node n, we may define the loss as the negative deviation from a target:

Ln = max[0, μWn − Wn], (2.14)

where μ is a user-defined percentage and Wn represents the capitalized initial wealth defined as295

Wn = W0(1 + γ )tn . (2.15)

Here, the initial wealth W0 is defined according to (2.13), γ is the per period interest rate and tn denotesthe time stage of node n.

The CVaR is defined as the expected value of losses exceeding VaR (denoted by ξβ) for a givenconfidence level β:

Φ = ξβ + 1

1 − β

∑n∈S

πn max[0, Ln − ξβ]. (2.16)

It is worthwhile observing that CVaR is a ‘coherent’ risk measure in the sense defined in Artzner et al.300

(1999), suitable for asymmetric distributions and thus able to control the downside risk exposure. Inaddition, it enjoys nice computational properties (see Ahmed, 2006) and admits a simple linear refor-mulation. In particular, CVaR can be linearized by introducing a set of additional variables (vn) andconstraints (2.17)–(2.19):

Φ = ξβ + 1

1 − β

∑n∈S

πnΔn, (2.17)305

Δn � Ln − ξβ ∀n ∈ S, (2.18)

Δn � 0 ∀n ∈ S. (2.19)

3. Solution approach

The proposed formulation belongs to the class of mixed-integer multistage stochastic programmingproblems. It is widely recognized (see, e.g. Birge & Louveaux, 1997) that the multistage paradigm pro-310

vides a very flexible modelling tool because it allows the dynamic and stochastic features characterizingmost of real-life applications to be dealt with jointly. Unfortunately, this modelling power comes at theexpense of an increased computational complexity. Indeed, the problem size grows exponentially withthe length of the time horizon. The decision variables are node dependent and their number can be hugefor a scenario tree of reasonable size. In addition, in the case of the capital budgeting problem, the deci-315

sion variables are binary. The resulting increased complexity may limit the adoption of general purpose


software for solving the deterministic equivalent mixed-integer problems calling for the definition ofspecialized solution methods.

Even if the literature findings on solution methods for mixed-integer multistage stochastic pro-gramming models are less than those for the two-stage counterpart, some contributions and research 320

streams are noteworthy. Decomposition approaches based on Lagrangian relaxation have been effi-ciently applied in the two-stage case (see Caroe & Schultz, 1999), but their suitability for multistagemodels is not completely proven. A Branch and Price approach has been proposed in Lulli & Sen (2004).The original problem is decomposed into a set of single-scenario subproblems, coordinated by a mas-ter. Lower bounds are obtained by column generation. A very interesting research stream is the Branch 325

and Fix coordination strategy proposed in Alonso-Ayuso et al. (2003) and also applied and extended inEscudero et al. (2010, 2012). This method, which relies on the twin-node family concept for satisfy-ing the non-anticipativity constraints explicitly imposed on the binary variables, allows also large-scaleproblems where uncertainty is represented via non-symmetric scenario trees to be dealt with.

In this paper, we propose to solve the multistage capital budgeting problem by enhancing the stan- 330

dard branch-and-bound method with specific branching rules which exploit the model structure. Asis known, the branch-and-bound method relies on an implicit enumeration of the search space. Thealgorithm starts by solving the continuous relaxation of the original problem. If one or more of thebinary variables has a fractional value, a variable is chosen for branching. In this way, two subproblemsare created and added to a list of active subproblems. At each iteration of the algorithm, a subproblem is 335

solved, providing an upper bound for the original solution. The enumeration at the current node of thesearch tree can be stopped if some conditions hold (the relaxation is infeasible, the optimal solution ofthe subproblem is not better than the incumbent value, the optimal solution of the integer subproblem isinteger feasible), otherwise two new subproblems are created and added to the list of active problems.The algorithm is iterated until the list of open subproblems is empty or a stop condition is satisfied. It is 340

evident that one of the key ingredients affecting the performance of the branch-and-bound algorithm isthe branching criterion. In what follows, we detail the branching rules adopted in the proposed solutionapproach.

3.1 The mean-risk branching rule

The mean-risk branching rule relies on a priority criterion defined by considering the specific problem 345

structure. We recall that in our model, binary variables are associated with both first-stage decisions xj

and staged decisions ykn. We observe that, because of the problem structure, if a variable xj is set to 0,the corresponding project is not accepted and, as a result, it can not be decommitted in later stages. Asa consequence, all the staged variables yjn can be fixed to 0. This leads to a reduction in the numberof decision variables equal to N , where N is the total number of nodes in the combined scenario tree. 350

Moreover, if a variable ykn is set to 1 then project k is terminated, and since this condition may occur onlyonce along a scenario, then all the staged variables related to project k and associated to the nodes of thesubtree rooted at n can be set to 0. This leads to a reduction in the number of variables equal to p(T−tn),where p represents the grade of node n, that is, the number of successors of node n (we assume thateach node has the same number of children) and tn denotes the time stage of node n. The considerations 355

introduced above are at the basis of the proposed branching rule. Indeed, the lower the level of thedecision variable considered for branching, the higher the reduction. Thus, our branching rule assignsthe highest priority to first-stage variables. In case of a tie, the choice is made according to the nature ofthe objective function. We recall that our model presents a mean-risk objective function. When the valueof λ is set to 0, risk is not accounted for. In this case, the decision maker exhibits a risk-neutral position 360

Antonio e Martina

Nota

it is known


and the model simply relies on the maximization of the expected value. In such a situation, we maydefine the priority weight according to the expected value (computed at the leaf nodes) associated witheach project. The intuition behind this rule (defined in the following as the M rule) is that the projectwith the largest value should produce the most significant impact on the portfolio expected value. Onthe contrary, when λ is set to 1, the decision maker is risk averse. In this case, the priority associated365

with each variable is determined by the quantity∑

n∈S πn max(0, Vj0 − Vjn). We observe that, becauseof the nature of the objective function, this value could be used as a proxy for risk. Selecting the variablewith the largest priority should, in this case, lead to the definition of two new subproblems with optimalvalues substantially different from those of the parent node. In the following, we shall refer to thissecond rule as the R rule. When the value of λ is within (0, 1), we have empirically found that for values370

close to 0 it is better to consider the M priority rule and the contrary when the λ value is close to 1. Turnnow to consider the case where the yjn are fractional. In this case, two different situations are worthconsidering. The first one occurs when, for a given project j, more than one variable associated withdifferent nodes n of the same level assumes a non-integer value. The second situation occurs when fora given node of the scenario tree, the decision variables associated with different projects are fractional.375

In both cases, we select as the candidate variable for branching the one with the most fractional valuein the linear relaxation.

4. Computational experiments

In this section, we report on the computational experience carried out with the twofold aim of validatingthe proposed multistage stochastic programming formulation and to preliminarily test the specialized380

branch-and-bound solution strategy. Regarding for the first aim we have considered a small but mean-ingful test case. As for the second, a set of randomly generated problems has been considered andsolved. All the problem instances have been generated in GAMS,3 a high-level modelling system formathematical programming and optimization, whereas the scenario tree generation procedure has beenimplemented in MATLAB 7.0.4 The branch-and-bound algorithm has been implemented in GAMS and385

uses the solver of ILOG CPLEX 12.2 with all internal parameters kept to default values. The numericalresults have been collected on an Intel Core 2 Duo CPU at 2.00 GHz and 3 GByte RAM.

4.1 An illustrative example

We consider the case of a company that wants to define its strategic plan over a time horizon of 6 timeperiods of 3 months each. We assume that there are no previously accepted projects (I = {}), whereas390

the set of projects J has cardinality 10. The set J is supposed to be partitioned into two subsets, eachcontaining projects which are perfectly correlated. The correlation between the two subsets is set to 0.5.

Table 1 depicts the main characteristics of each project. We suppose that the project’s initial valueVj0 and threshold Gj are equal to the required initial investments Fj. The company has an initial budgetof 400K e and additional resources for 50K, 40K, 30K, 20K and 10K e for the following time periods,395

respectively. We also consider the possibility of investing the extra cash flow at a constant risk-free rateof 1%, while we fix the parameter μ equal to 1. The drift term for each project has been set to 1%.

The uncertain evolution of the project’s value is represented by a scenario tree defined by the proce-dure illustrated in Section 2.1.1. Part of the tree is depicted in Fig. 3 where we report the project values,but we omit the cash flows since they can be easily derived starting from the project value.400

3 www.gams.com4 www.mathworks.com


Table 1 Projects’ characteristics

1 2 3 4 5 6 7 8 9 10

Initial value (Vj0) 110 120 200 300 200 200 160 180 200 140Recovery rate (Rj) [%] 20 30 20 50 30 20 30 20 50 30Volatility (σj) 0.2 0.1 0.2 0.1 0.2 0.3 0.2 0.3 0.1 0.2

Fig. 3. Part of the scenario tree with project values.

Figure 4 shows the recommendations provided by the multistage solution for a given scenario whoseproject values are depicted in Fig. 3. As is evident, the model initially suggests selecting projects 1, 2, 9and 10, but because of adverse conditions, some projects are (under this scenario) terminated early. Thisis because of the possibility of disinvesting from projects which are considered to be not so profitable forthe residual time horizon and invest the available resources in the risk-free asset. We underline that the 405

possibility of taking corrective action represents the strength of the multistage stochastic programmingapproach which, in contrast to the deterministic counterpart, allows a more flexible and active strategyto be implemented.

We further observe that the proposed mathematical formulation provides the decision maker with atool to assess risk aversion. Figure 5 shows the efficient frontier representing the set of non-dominated 410


Fig. 4. Recommendations provided by the model.

solutions obtained for different values of the λ parameter. For each solution, there is no other strategywhich is contemporaneously more profitable and less risky. The marked points have been obtainedby solving the stochastic model with the λ parameter varying from 0.99 to 0.01. As is evident, thehigher the λ value, the less risky (and less profitable) the investment strategy. This analysis allows thedecision maker to choose the best solution in terms of return/risk trade-off according to requirements415

and strategies.

4.2 Approach evaluation

The solution of a stochastic programming formulation comes at the expense of higher com-putational requirements. Thus, the first observation that can be made is whether the recourseto a more advanced modelling framework is really worthwhile. With the aim of assessing this420

important issue, different numerical experiments have been carried out. The first one is thecomparison between the solutions provided by the stochastic formulation and its determinis-tic counterpart. We have measured the so-called Value of Stochastic Solution (VSS) that is astandard measure of the effectiveness of a stochastic programming model (see Birge & Lou-veaux, 1997). We have built up a deterministic model obtained by considering for each stage425

the expected value of input parameters, i.e. the expected value computed on the cash-flow


Fig. 5. Efficient frontier.

and project values associated to nodes of the same level, and used this model as a benchmark for thestochastic one.

The VSS measure was originally proposed for two-stage models, but recently it has been extendedfor multistage formulations (see Escudero et al., 2007; Vespucci et al., 2010). The authors have proposeda procedure for the calculation of a ‘dynamic’ solution of the deterministic model based on the average 430

scenario, thus providing a better approximation for the stochastic model than the classical deterministicaverage scenario model.

We have adopted this framework to evaluate the advantage of a stochastic approach, obtaining avalue of 367.5K e for the dynamic VSS. This value is about 29% of the objective function value ofthe proposed model, showing a clear advantage in the use of a stochastic framework with respect to a 435

deterministic one.Numerous experiments have been carried out with the aim of simulating the use of the proposed

model in a real setting. It is evident that in current practice, decision makers are called to periodicallymonitor the evolution of the defined investment plans, introducing some changes if the real observedevolution differs from the target one. This important dynamic feature of the problem can be mathe- 440

matically translated by including the proposed formulation in a rolling horizon framework. We observethat, over the planning horizon, the investment strategy actually implemented by the decision makerconsiders a sequence of first-stage (here-and-now) decisions associated with available information. Inparticular, at each stage, as uncertainty unveils, a new scenario tree can be generated and a new decisionmodel can be formulated and solved over a moving rolling time window: as time evolves every imple- 445

mented decision will provide the best hedged plan against the set of scenarios starting from the currentroot node.

A possible iterative schema for this policy is the following.

• Fix strategic goals (planning horizon, budget, macro-areas).


• For each time period t of the planning horizon,450

◦ define the current universe of investable projects and the set of existing projects;

◦ update information about current and historical projects’ values;

◦ generate up-to-date uncertainty representation and determine the optimal project portfolio;

◦ implement the here-and-now decisions.

The rolling framework approach has been applied to further evaluate the real advantage achievable455

by the application of the stochastic programming approach. To this aim, a comparison with other deci-sion strategies has been performed. We have considered as a benchmark two simple approaches: thedeterministic model as defined above and a ‘static’ model in which project-related decisions are pos-sible only at the first stage without any recourse decision for subsequent periods. The comparison hasbeen performed on an out-of-sample basis. In particular, since no historical information on a project’s460

evolution is available, a ‘quasi’ real-life evolution has been simulated by means of a scenario built in asimilar way with respect to the adopted scenario tree. In particular, this scenario has been generated bymeans of a Monte Carlo procedure based on the same evolution model specified in Section 2.1, with anadditional standard normal random component. This scenario, different from the ones included in thescenario tree, has been considered as the real project evolution over the planning horizon.465

The simulation has been carried out in a rolling fashion: after an initial project portfolio definitionfor each one of the strategies at hand, at any stage of the planning horizon, information on projectvalues and cash flow is updated, uncertain future projects’ evolution is eventually modified (by meansof a new scenario tree) and a new project portfolio is defined. As a performance index over the planninghorizon, we have considered the return obtained by each of the investment strategy. This value considers470

the portfolio wealth cleaned by initial budget and subsequent available resources. Figure 6 reports thereturn values for each period of the planning horizon of the three strategies at hand (deterministic, static,multistage). The multistage approach clearly outperforms the others. Except for one stage (the third), atwhich we register similar performances, the multistage stochastic programming strategy allows a higherreturn in each period of the planning horizon. The difference with the static strategy is due to the fact475

that the stochastic dynamic approach suggests initial decisions which not only take into account severalmarket evolutions but also hedge against future adverse conditions that may occur.

4.3 Numerical results

The prototype branch-and-bound method introduced in Section 3 has been preliminarily tested on a setof randomly generated test problems. In particular, different instances have been generated by varying480

the number of projects (NP), the size of the scenario tree in terms of node number (NN) and the riskaversion parameter (RL). Thus, each instance is defined by (NP.NN.RL). A set of 16 test problems hasbeen randomly generated by setting the number of projects to 10, 20, 30 and 50, the number of nodes to1,365 and 5,461, and the risk aversion parameter to either a low (l) or high (h) level. In each instance,the whole project set is partitioned into two correlated subsets of equal size (the projects within each485

subset have correlation 1). Table 2 reports for each test problem the dimension in terms of projects,scenarios, nodes of the scenario tree, constraints, integer and continuous variables, constraint matrixnon-zero elements and density.

The performance of the branch-and-bound algorithm in terms of solution time in seconds for boththe standard CPLEX branching rule (with default options) and the proposed one is reported in Tables 3490


Fig. 6. Return comparison of deterministic (solid line), static (dotted line) and dynamic (dashed line) strategies.

Table 2 Test problem size

Test problem NP Scen. NN Constr. Total var. 0–1 var. Non-zero elements Density

10.1365.h(.l) 10 1,024 1,365 17,745 17,746 13,650 67,592 2.15 × 10−4

10.5461.h(.l) 10 4,096 5,461 70,993 70,994 54,610 286,830 5.69 × 10−5

20.1365.h(.l) 20 1,024 1,365 31,395 31,396 27,300 122,900 1.25 × 10−4

20.5461.h(.l) 20 4,096 5,461 125,603 125,604 109,220 573,460 3.63 × 10−5

30.1365.h(.l) 30 1,024 1,365 45,045 45,046 40,950 184,350 9.09 × 10−5

30.5461.h(.l) 30 4,096 5,461 180,213 180,214 163,830 860,190 2.65 × 10−5

50.1365.h(.l) 50 1,024 1,365 72,345 72,346 68,250 307,250 5.87 × 10−5

50.5461.h(.l) 50 4,096 5,461 289,433 289,434 273,050 1,310,074 1.56 × 10−5

and 4. A limit of 5000 s has been imposed on the CPLEX solution time. As expected, the solution timegrows with the number of nodes of the scenario tree. This is motivated by the nature of the decisionvariables that are node dependent.

The results clearly show that the proposed branching rules allow a better computational efficiencythan the standard one. As is evident, a good percentage reduction (around 45%) in the computing time 495

has been registered, together with a reduction in the number of branch and bound nodes. Similar perfor-mances have been observed for all the test cases.

The analysis of these preliminary results clearly underlines the effect produced by the adoption ofthe proposed branching rule on the solution method. In spite of its simplicity, the proposed branchingrule seems to be quite effective, producing a significant reduction in the required solution time. 500


Table 3 Numerical results for high values of risk aversion parameter

Default BR M(R) BRTest problem Branching nodes Solution time (s) Branching nodes Solution time (s)

10.1365.h 6 43.25 4 12.0010.5461.h 33 334.13 7 72.77

20.1365.h 222 278.60 110 156.0020.5461.h 649 2,164.00 407 1,447.90

30.1365.h 1,119 622.09 609 358.3030.5461.h >7,132 >5,000 5,065 2,807.00

50.1365.h 1,964 1786.31 895 895.4050.5461.h >4,350 >5,000 3,599 3,917.00

Table 4 Numerical results for low values of risk aversion parameter

Default BR M(R) BRTest problem Branching nodes Solution time (s) Branching nodes Solution time (s)

10.1365.l 7 41.38 4 12.1410.5461.l 34 338.30 8 73.19

20.1365.l 219 246.67 107 143.0220.5461.l 671 2,421.30 389 1,254.46

30.1365.l 1,142 642.43 638 416.4930.5461.l >7,148 >5,000 5,843 3,047.84

50.1365.l 1,899 1,681.27 812 752.6750.5461.l >4,382 >5000 4,009 4,140.66

5. Concluding remarks

In this paper, we have addressed the capital budgeting problem under uncertainty using the multi-stage stochastic programming framework. With respect to more traditional approaches, the multistageparadigm allows the decision maker to define more flexible management plans, by modifying previouslydefined project portfolios if unfavourable conditions prevail. Risk is explicitly assessed by defining a505

mean-risk objective function, where the CVaR on the potential losses is used. The uncertain evolu-tion of the project value has been represented with a scenario tree, generated using the Amin (1991)approach. The proposed model belongs to the class of integer multistage problems. Because of size, itssolution may be a very critical issue. For this reason, a customized branch-and-bound method basedon the mean-risk branching rule has been proposed and implemented. Extensive computational experi-510

ments have been carried out to validate the model’s effectiveness, also in comparison with other possiblebenchmark policies. The numerical results collected by solving randomly generated instances with theproposed branch-and-bound approach seems to be very encouraging.

However, since a significant increase in the number of projects, stages or scenarios could make real-life instances of the proposed model very difficult from a computational standpoint, we are working on515

the inclusion of our branching idea within decomposition approaches.


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