multi-criteria project portfolio optimization under … multi-criteria... · multi-criteria utility...

Business Administration and Management

714, XVI, 2013

Introduction

In reality we can often come across the task ofdeveloping a portfolio (set) of projects havingthe character of investment projects(extending production capacity by means ofdevelopment or acquisitions, innovation ofproduction technologies, introducing newproducts and such like) or research projects(research and development of new products,technologies, processes and such like).Creating these portfolios for the individualgroups of projects represents a base forconceiving an investment programme ora research programme of a company. Creatingportfolios of projects has its own processaspects as well as model aspects. Theprocess aspects consist in the desired processof the project portfolio development from thepoint of view of the individual steps and theircontent [7]. The model aspects, which are dealtwith in this contribution, relate to applyingcertain optimization models supporting thecreation of portfolios. With regard to theuncertainty of many factors influencing thefuture projects results these models are to beperceived as stochastic optimization models.

The aim of the contribution is to brieflycharacterize the types of the tasks concerningthe optimization of a portfolio, to specify thenature of the development of a project portfoliounder risk and to show the solution of this task,partly by means of the deterministic equivalentsof the stochastic optimization tasks and partlyas an application of the optimization toolOptQuest (add-in MS Excel Crystal Ball) for thestochastic optimization. Both these optimizationapproaches are illustrated on practical examplesdemonstrating the types of information that can

be gained for increasing the quality ofinvestment plans and decision making.

1. Classification of the PortfolioOptimization Tasks

The issue of the development, or, as the casemay be, the optimization of a project portfoliounder risk belongs to a broad group of theportfolio optimization tasks. This group can beclassified according to a number of viewpoints,namely those as follows:� the character of variables, which may be

either continuous or discrete. The firstworks from the field of the portfolio theorydealt with the allocation of financialinvestment with continuous variables andthey are associated with the name of H.Markowitz [23]. Only later attention waspaid to the tasks with the discrete orbivalent variables, and these represent thecore of this contribution;

� different level of uncertainty concerningparameters of the portfolio developmenttask, when on the one hand availableinformation makes it possible to reach thedetermination of their divisions of pro-bability, on the other side it is impossible todetermine even the state of the worldwhereon the values of these parametersdepends ([2], [3], [11]);

� the number of assessment criteria, whenthe tasks may be of the mono-criteria andmulti-criteria nature. A larger number ofworks from the field of portfolio developmenthas a mono-criteria character and theseworks apply various approaches. Theseare, for example, the portfolio optimizationby means of mathematical fuzzy logicprogramming [13], the portfolio optimization

MULTI-CRITERIA PROJECT PORTFOLIOOPTIMIZATION UNDER RISK AND SPECIFICLIMITATIONSJifií Fotr, Miroslav Plevn˘, Lenka ·vecová, Emil Vacík

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with stochastic yields by means of fuzzyinformation [16], and the choice ofa portfolio under stringent uncertainty [3].As far as multi- criteria oriented works areconcerned, it is possible to mention forexample the choice of a portfolio by meansof multi-criteria stochastic programming [1]or optimization of investment in theselected fields of the national economybased on the method of linking scenarioswith the method of the multi-criterialimitation PROMETHEE [15];

� variables of the model of portfoliodevelopment which have the nature ofrandom quantity. These may become thecoefficients of a criteria function, thecoefficients of variables under limitingconditions or the right sides of thoselimitations. The tasks with randomlyvariable coefficients of criteria functions arethe most frequent and relatively simpler;

� the nature of the solution of a model ofa portfolio development where thesubject matter may be optimization ora certain kind of reduction of the initial setof the objects from which a portfolio is beingcreated. As far as optimization is concer-ned, these tasks may be solved analyticallyby means of stochastic programming [29]supported by heuristic or geneticoptimization algorithms based on the MonteCarlo simulation [24], [14] or by trans-formation to the deterministic equivalents ofthe tasks of stochastic programming. Ofspecific nature are the tasks based on themodel (rule) of the mean value – scatterenabling to achieve effective portfoliosmaximizing the mean value of the yieldcriterion when the criterion is variablylimited to the risk measured by scattering orstandard deviation of this criterion (or if youlike, minimizing the risk of portfolio whenthis criterion is variably limited to its yield).This approach is again based on the worksof Markowitz (for more detail see, forexample, [34]). The reduction of a portfolioleading to the determination of an effectiveset may be, again, based on the model(rule) of the mean value – scatter, or on therules of the stochastic domination [10].In this contribution the authors try to

concentrate on the task of the optimization ofa project portfolio under risk, partly by the

solution of the deterministic equivalents of thistask in the form of multi-criteria or mono-criteriabivalent programming under uncertainty in theassessment criteria, and partly by applying theoptimization tool OptQuest for the stochasticoptimization with continuous or discretevariables, based on linking the heuristicoptimization algorithm with the Monte Carlosimulation [4], [25].

2. The Character of the Task ofCreating a Project Portfolio

The core of the task of creating a projectportfolio is creation of such a set of projects, outof a set of prepared projects, which meetscertain conditions. The implementation of eachproject requires applying certain sources andthat is why the task of creating a portfolio is alsothe task of allocation of (usually scarce)sources. If, in the process of creating a portfolio,our aim is maximization or minimization ofcertain characteristics of a portfolio, the taskcan be described as optimization in the form ofan optimum allocation of sources.

The creation of a project portfolio hasusually certain common features, namely thefollowing ones:� The multi-criteria character of the task,

because more objectives are to be met andthe level of accomplishment is expressed bymeans of the individual assessing criteria.

� Uncertainty of some factors influencing theresults of the projects and therefore itssuccess and hence the risky nature of theprojects.

� Scarcity of sources meaning the fact thatindividual projects should not beconsidered in isolation as the acceptance ofa certain project decreases disposablesources meant for other projects. Thisscarcity of sources calls for the need ofusing optimization tools.

� Mutual dependency of projects has to betaken into account too. The dependency ofa portfolio can have the character ofstatistical dependency (direct or indirectdependency of various intensity expressedby means of correlation coefficients of thepairs of investment projects in the form ofrandom quantities) or the character offunctional dependency (e.g. a certain projectcan be put in a portfolio only if anotherparticular project has also been put there).

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3. The Optimization of a ProjectPortfolio under Risk by Means ofthe Deterministic Equivalents ofthe Task of StochasticProgramming

The deterministic equivalents of the tasks of thestochastic optimization transform these tasks tothose of the deterministic type, when thedivisions of the probability of the stochasticvariables of the model are replaced by theirstatistical characteristics in the form of themean values and scatterings.

In the following section (3.1) we will try andcharacterize the solution of the deterministictask of the multi-criteria optimization withrandomly variable criteria by means of themulti-criteria utility function [18] as a tool of themulti-criteria assessment of the projects underrisk, and also by means of the programmeLingo (for the solution of the tasks of linear andnon-linear bivalent programming) [17]. In thenext section (3.2) we will apply a similar approachby means of the mono-criteria optimization ofa project portfolio under risk, and in the lastsection (3.3) we will try to characterize brieflyother types of the deterministic equivalents ofthe tasks of the optimization of a portfolio underrisk.

3.1 Multi-criteria Project PortfolioOptimization under Risk withRandomly Variable Criteria ofAssessment

Task formulationLet us assume that n projects have beenprepared for the development of a companyinvestment programme out of which it isnecessary to draw up a portfolio maximizing itsmean utility value from the point of view ofm criteria when respecting p scarce sources.For each project there are divisions ofprobability of the individual assessment criteriaavailable and their statistical characteristics inthe form of the mean value and the standarddeviation, the source demandingness and thedisposable volumes of each source. (We canuse the Monte Carlo simulation to determinethe probability divisions of the quantitativecriteria – for more detail see e.g. [8], [14] or[24], [25].)

Now, the deterministic equivalents of thetask of the project portfolio optimization can be

formulated as a task of bivalent programmingwith the criteria formulation based on the conceptof multi-criteria utility function under risk anda set of source limitations. The construction ofthe multi-criteria utility function under risk is noteasy and it requires the following:� to verify preferential and utility

independence of the criteria (only in thiscase it is possible to express the multi-criteria utility function under risk in theadditive or multiplicative form);

� to construct partial utility functions ui(xi) ofthe individual criteria and determine itsweight in the dialogue of an analyst witha decision maker (for more detail see e.g.[6] and [18]),where the variable xi expressesthe consequence of the given project withregard to the i-th assessment criterion;

� to specify the form of the multi-criteria utilityfunction under risk.In case the requirement for the preferential

and utility independence is met and the sum ofthe weights of the assessment criteria equalsone (or at least approximately equals one), it ispossible to express a multi-criteria utilityfunction under risk in an additive form by therelation (1). In the opposite case it is necessaryto express the multi-criteria utility functionsunder risk in a more complicated multiplicativeform (for more detail see again [6] and [18]).

(1)

In case of the existence of more projectsthe utility of a j-th project U(Xj) can beexpressed by means of the relation (2):

(2)

where Xj – vector of the consequence of j-th

project with regard to the individualassessment criteria,

xji – consequence of j-th project with

regard to i-th assessment criterion(i =1, 2,…,m and j =1, 2,…,n),

ui(xji) – utility of i-th consequence of j-th

project,vi – weight of i-th criterion.

From the knowledge of the partial utilityfunctions of the individual criteria and theirdivision of probability we can derive the mean

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utility values E[ui(xji)] of each project with regard

to i-th criterion. For a continuous criterion weuse the relation (3):

(3)

whereFj

i(xji) – distribution fiction of j-th project with

regard to i-th criterion,di – lower limit of the definition domain of

a partial utility function of the i-thcriterion,

hi – upper limit of the definition domain ofa partial utility function of the i-thcriterion.

In case of the assessment criteria ofdiscrete nature the mean values of partialutilities can be set more easily as a sum ofpartial utilities weighed by their probabilities.

The knowledge of the mean values ofpartial utilities enables (in case of the additiveform of the multi-criteria utility function) to setthe mean value of the overall utility of the j-thproject E[U(Xj)] in the form (4):

(4)

The mean value of the overall utility of theproject portfolio can be now expressed easilyas a sum of the mean values of the overallutilities of the individual projects contained inthe portfolio. Let us define the bivalentvariables:

yi ∈ {0; 1} gains the value 1 (j-th project isput to the portfolio) or 0 (j-th project is not put tothe portfolio).

The task of the optimization of a projectportfolio under risk can then be formulated asa task of bivalent programming with a criteriafunction expressing the mean value of theoverall utility of the portfolio in the form (5):

(5)

and with a set of source limitations asformulated in (6):

(6)

whereaj

k – consumption of k-th source for j-thproject (k = 1, 2,…, p),

Lk – disposable volume of k-th source.

The model of bivalent programming can beextended further by limitations requiring theachievement of at least certain values of somequantities (let us indicate the set of the indexesof the quantities as Q) representing furtherrequirements that the portfolio should meet(e.g. the requirement for achieving a certainsize of sales, not exceeding the given risk andsuch like). The respective limitations foradditive quantities of the yield type, amongwhich there may also be some assessmentcriteria, might have the following form:

(7)

whereVq

(min) – the lower limit of q-th quantity of theyield type of the project portfolio,

hji – the value of i-th quantity of the yield

type of the j-th project (these are wellknown constants).

In case of the opposite requirement, i.e.non exceeding of the given values of certainquantities the relevant limitations should havean analogical form as the source limitations ofthe model of the bivalent programming.

An Example of the Solution of a Multi-criteriaProject Portfolio Optimization under RiskLet us demonstrate the solution of a task ofmulti-criteria project portfolio optimization underrisk on an example of the project focused onintroducing new products. Let us assume thatfor the development of this portfolio there are12 projects available which were assessed withregard to the set of five criteria formed by twoquantitative criteria (net present value (NPV)and cost effectiveness of the capital) and threequalitative criteria (concordance with thecompany strategy, market attractiveness andsupport of key competencies). The higherweight among the economic criteria went to theNPV; weights of the noneconomic criteria werejudged. The division of the probability of thequantitative criteria for the individual projectswas set by the Monte Carlo simulation; thedivision of probability of the qualitative criteria

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(transformed to discrete quantitative criteria)was done in an expert way. For each of theassessment criteria a partial utility function wasspecified. On the basis of these functions andthe relevant divisions of probability as well asaccording to the relation (3), with continuouscriteria or with analogical summative relation ofdiscrete criteria, mean utility values were set foreach project with regard to the individualcriteria. By means of the relation (4) the meanvalues of the overall utilities were set.

The optimization of the project portfolio hadalso to respect two limited sources, namely thecapital budget (CZK 560 million) and a dispo-sable number of workers (240). The demands

of the individual projects as far as the workerswere concerned were set with a significant levelof reliability and that is why it was possible towork with them as with deterministic quantities.The estimates of investment costs of theindividual projects were considerably uncertainand that is why they were represented byrandom quantities with expertly set subjectivedivisions of probability. With regard to the factthat NPV represents a key financial criterion ofthe project assessment, even the calculation ofthe portfolio NPV in the form of its mean valuewas part of the optimization project (a sum ofquantities for the project portfolio optimizationis summarized in Table 1).

Tab. 1: Characteristics of Investment Projects

Project E(U)E(NPV) E(IN) Number

(mil CZK) (mil CZK) of workers

Project 1 0.75 22.30 31.0 23 2.42

Project 2 0.36 5.16 27.6 12 1.30

Project 3 0.86 28.21 126.8 45 0.68

Project 4 0.59 26.32 96.5 37 0.61

Project 5 0.75 15.32 55.8 25 1.34

Project 6 0.78 10.47 36.8 23 2.12

Project 7 0.45 12.30 44.7 15 1.01

Project 8 0.65 23.01 67.5 26 0.96

Project 9 0.76 15.07 49.0 24 1.55

Project 10 0.48 29.47 85.8 38 0.56

Project 11 0.52 20.24 53.3 20 0.98

Project 12 0.74 12.35 42.3 17 1.75

Note: E(U) – mean value of the project overall utility,E(NPV) – mean NPV value for a given project,E(IN) – mean value of the investment costs of a given project,

– mean value of a project utility related to 100 million of investment costs spent (IN) expressed again by their mean value.

Source: own calculations

E(U)––––––––E(IN100)

E(U)––––––––E(IN100)

If all the prepared projects were put in theportfolio, the limited sources would beexceeded considerably. (The demandingnessof the implementation of the portfolio on thefunds would amount CZK 717.1 million, whichmeans exceeding the limit of CZK 560 millionby CZK 157.1 million and the disposablenumber of workers of 240 would be exceededby 65; the mean utility value of this portfolio

would be 7.69 and the mean NPV value wouldbe CZK 220.2 million.) Therefore a deterministicequivalent of optimization of a project portfoliounder risk was used for the development of theportfolio, which has a form of a bivalentprogramming model. The criteria function ofthis model maximizing the mean utility value ofthe portfolio is given by the relation (5) and itstwo source limitations relating to the capital

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budget and the disposable number of workersare given by the relation (6). By solving theoptimization task by means of the software toolfor bivalent linear and non-linear programmingLINGO (see for example [17]) we gained anoptimum portfolio created by projects 1, 3 up to6, 8, 9, 11 and 12. The mean utility value of thisportfolio was 6.40, the mean NPV value wasCZK 173.3 million and almost all the budgetwas drawn (only CZK 1 million remaineddisposable) and all 240 workers were fully used(see the solution in Table 2 for the portfolio A).

Another subject of interest consisted infinding out what impacts an increase of therequirement concerning the mean NPV valuewould have on the optimum portfolio. Two tasksof bivalent programming were gradually solvedwith the requirement to achieve the mean NPVvalue of at least CZK 175, or, as the case maybe, CZK 180 million. The results of both theseoptimizations (portfolio B and portfolio C)together with the results of optimization withoutthe limitation to the mean value NPV (portfolioA) are summarized in Table 2.

Tab. 2: The Results of the Portfolio Optimization

Portfolio Portfolio structure E E Utility decrease NPV growth Source demands

(U) (NPV) abs. % mil CZK % F W

A 1, 3–6, 8, 9, 11, 12 6.40 173.3 559 240

B 1, 2, 4, 6–12 6.08 176.7 0.32 5 3.4 2.0 535 235

C 1, 2, 4, 5, 7–12 6.05 181.5 0.03 0.5 4.8 2.7 554 237

Note: E(U) – mean utility value of the project portfolio,E(NPV) – mean NPV value of the project portfolio,F – funds,W – number of workers. Source: own calculations

As is obvious from this table the require-ment for the mean value to achieve at leastCZK 175 million (portfolio B) leads to thedecrease of the mean utility value by 0.32 (5 %),while the growth of the mean NPV value is onlyCZK 3.4 million (2 %). Further tightening of therequirement for NPV (achieving at least CZK180 million, portfolio C) leads to a considerablesmaller decrease of the mean utility value(absolutely 0.03 or 0.5 %) as opposed toportfolio B, while the growth of the mean valueNPV is now higher (CZK 4.8 million or 2.7 %).

The results of the optimization calculationscan now serve as a valuable source of infor-mation of a company management for thedecision about the structure of the portfolio to beimplemented. In this decision making process itis also necessary to consider other aspects notincluded in the optimization, including thechange in drawing the limited sources (e.g. theidle CZK 25 million with portfolio B).

The optimization of the project portfoliocould be, to a certain extent, simplified in ourtask in case of the existence of only one limitedsource. If this was the capital budget, we coulduse the indicator E(U) / E(IN100) whose values

for all projects are stated in Table 1. If weordered the projects according to the decreasingvalues of this indicator and gradually put theprojects according to the decreasing values ofthis indicator down to achieving the capital limit,we would, in this simple way, proceed toa relatively good portfolio. In case of therequirement for finding an optimum portfoliothis task may lead to the solution of the taskcalled a knapsack problem, which is a relativelysimple task of integer linear programming. Ifwe, in our case, applied the above simpleprocedure, the projects 4 and 10, i.e. those withthe lowest mean utility values at CZK 100million of investment costs, would not get anyfunding. This way of optimization of the researchof the new medicaments has been applied byone of the world’s biggest pharmaceuticalcompanies GlaxoSmithkline. The mean NPVvalue on USD 1 million serves as theoptimization criterion (for more detail see [30]).

What is seen as a considerable disadvan-tage of the multi-criteria project portfolio optimi-zation under risk is the difficulty of constructingpartial utility functions, which managers do notlike to work with. Applying the Cost Benefit

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Analysis (CBA) – (see [5]) may be a solution asin some cases this method enables a transferof criteria values expressed in the non-monetary units to a monetary expression andthis way the monetary assessment can beintegrated for example to the indicator of thenet present value (NPV) which might serve asan overall criterion of the project portfoliooptimization. Another possibility is the transferof certain criteria of assessment to the limitationconditions, i.e. especially those criteria whichwere not possible to transform to a monetaryexpression. This procedure is not complicatedas it only leads to increasing the number oflimitations of the optimization model for which itis necessary to set their right sides in the formof the lower limits (for the criteria of the yieldtype) or the upper limits (for the criteria of thecost type). In practice and sometimes even inprofessional literature (for example [22]), wecan come across attitudes where even the projectrisk is included in the assessment criteria. Themain drawback of this attitude based on themulti-criteria project assessment as if carriedout in the conditions of certainty (i.e. on the basisof only one scenario) is that it does not respectthe dependency of the project assessment withregard to some criteria under this type of risk(a higher risk only decreases the overallassessment of projects).

3.2 Mono-criteria Project PortfolioOptimization under Risk witha Randomly VariableAssessment Criterion

The model of bivalent programming for themono-criteria optimization of a project portfoliounder risk can be easily derived from the modelof the multi-criteria assessment. Let us assumethat a key additive optimization criterion c of theyield type has been chosen. Further, weassume that the mean values E(cj) for theindividual projects (j = 1, 2, …, n) are known andthe bivalent variables are defined as yj∈{0; 1},where for yj = 1 j-th project is put to portfolioand for yj = 0 is not included in the portfolio.Then the criterion function of the deterministicequivalent of the stochastic optimization ofdevelopment of a project portfolio under riskcan be expressed as (8):

(8)

The set of the source limitations of themodel or the limitations expressing therequirements for non-exceeding of certainvalues of the chosen quantities of the cost typemodel can be expressed by means of therelation (6). The same way the requirements forachieving at least certain values of the chosenquantities of the yield type model can beexpressed by means of the relation (7).

In the tasks concerning the project portfoliooptimization under risk, we have not yet, withinthe set of the model limiting conditions,included the limitation concerning the risk ofa project portfolio expressed in relation to theoptimization criterion. The optimal portfoliomaximizing (in case of the yield type criterion)the mean value of this portfolio might beconsiderably risky as the more risky projectsusually lead to higher yields. With regard to thisit is necessary to extend the set of limitations ofthe bivalent programming model by thelimitation of the portfolio risk in the form of (9):

(9)whereσ – standard deviation of the optimization

criterion of the project portfolio,σi, σj – standard deviation of the optimization

criterion of the i-th or j-th project,σh – upper, limit of the portfolio risk with

regard to the set-down optimizationcriterion as expressed by the standarddeviation,

rij – correlation coefficient of the value of i-th or j-th project (these values can beestimated expertly on the basis of theproximity or dissimilarity of the subjectmatter of the projects, their marketdetermination, raw material base andsuch like, while a role is also played bythe relations of the systematic andspecific risks of the individual projects).

It is obvious that the less strict thelimitations (9) of risks are, the higher the meanvalue of the portfolio is. It might now be usefulto find out about the impacts of the tighteningup of the requirements for the portfolio risk asfor the structure and characteristics of the set-down optimal (effective) portfolios. The graphicdepiction of these portfolios then forms theefficient frontier (see below).

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3.3 Further Types of DeterministicEquivalents of the Tasks ofa Project Portfolio Optimizationunder Risk

In this section we are going to outline furthertypes of the tasks of the optimization ofa project portfolio under risk, partly bymaximizing the probability of exceeding thetarget value of the criterion, and partly by a taskwith randomly variable limitations.

Maximization of the Probability ofExceeding the Target Value of a CriterionLet us assume that NPV is the criterion of theproject portfolio assessment and its target(planned) value NPVC has been set. Oncondition that the variables yj are identical withthose in the text above, the criteria function ofthe project portfolio under risk can now bewritten down in the form of (10):

(10)

whereNPVj – the net present value of j-th project

(j =1,2,...,n),NPVC– target value of the net present value of

the project portfolio.

The limiting conditions expressing thesource settings are possible to express by therelation (6) and the limitations relating to therisk of the portfolio by the relation (9). In casethat NPVj can be expressed as a linear functionof a random parameter δ∈⟨0; 1⟩ in the form of(11):

(11)

the deterministic equivalent of the task of maxi-mization of the probability of exceeding the targetvalue NPVC of the portfolio can then be trans-formed to a task of linear fractional programmingwith a criteria function in the form (12):

(12)

with limitations (6) and (11). A solution of thistask using simulation is given in section 3.3.

Optimization of the Project Portfolio underRisk with Randomly Variable LimitationsThe following factors can be the randomlyvariables in this case:� the right sides of the limiting conditions

(usually of source limitations), and� the coefficients of the variables on the left

sides of the limitations (in case of sourcelimitations it is the demandingness of theindividual projects on the limited sources).The tasks with random variables on the

right sides of the limitations are easier to solve.In case of the source limitations it may be anexample of export activities from a remoteforeign country, raw material in the initial stageof extraction where a geological survey ofa deposit thickness has not been completed yetand such like.

If, for example, the disposable volume of k-th limited source is uncertain and we know itsdivision of probability, then a part of the relatedoptimization model will be for examplea limitation requiring that the probability that theconsumption of k-th source will not exceed itsmean value will be higher than the set value.An appropriate limitation should then have theform of (13):

(13)

whereαj

k – consumption of k-th source for j-thproject,

E(Lk) – mean value of the disposable volumeof k-th source,

αk – lower limit of the probability of non-exceeding the mean value of thedisposable volume of k-th source inpercentage.

For the deterministic equivalents of the taskof optimization of the project portfolio it is nowpossible to transform the above statedlimitation (see [12]) to the form (14):

(14)

whereL*k – the (1 – αk) percentile of the division of

the probability of disposable volume ofthe k-th limited source.

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In the tasks of the optimization of a projectportfolio under risk we can come across theuncertainty of the coefficients αkj of certainlimitations of the model more often. Thelimitation of the capital budget of a portfolio canbe a typical example of this, when theinvestment costs of the individual projects areconsiderably uncertain. The experience frombusiness practice shows that estimates ofthese costs are considerably optimistic andthey are usually exceeded – see [9]. If boththese coefficients and the right sides of therelevant limitations (volumes of disposableincomes) are random, then it is possible totransform them to the set of non-linearlimitations, and so the deterministic equivalentof the optimization of a project portfolio underrisk has the form of a model of non-linearbivalent programming [12].

4. The Application of the MonteCarlo Simulation in the Processof the Stochastic Optimization ofa Project Portfolio

As we have already stated, certain simplertasks of the optimization of the project portfoliounder risk are possible to solve by their transferto deterministic equivalents. Difficulties arise incase of more difficult tasks (not only thecoefficients of criteria function are of a randomnature but also the coefficients of the left sidesof the limiting conditions, and their right sides),or, as the case may be, in other types of thecriteria functions than the functions expressedby relations (5) and (8). Analytic solution ofthese optimization tasks of stochastic bivalentprogramming is usually considerably difficult,but the above mentioned optimization programscan be applied successfully.

Gradually four tasks of stochastic optimi-zation of the project portfolio were solved basedon an example whose deterministic equivalentwas solved in section 3.1. In this model thereare, on the whole, 24 random variables whichrepresent NPV and the investment costs foreach model. The division of the probability ofNPV of the individual projects were set by theMonte Carlo simulation, and these divisionswere approximated by the most suitable typesof the theoretical divisions (with five projects itwas normal divisions, with the remaining sevenprojects it was beta division with negativeskewness. The division of the investment costs

probability has mostly the character of betaPERTdivision (with expertly estimated parameters byauthors of this article) and positive skewness.As a basis of determination of the probabilitydivision of these costs it is possible to use thepost-audit results of similar projects that arefocused on detection and evaluation of deviationsbetween planned and for real expended investmentcosts. For more detail see [9] and [31].

With regard to the fact that the investmentcosts are one of the significant factors influencingNPV of each project, statistical dependency ofNPV and the costs were respected for eachproject (the expertly estimated correlationcoefficients on the basis of the investmentcosts contributions to the uncertainty of NPVwere set by the disperse analysis and theywere from -0.2 to -0.4). The model of this taskis given by Table 3. First we paid attention tothe optimization of the portfolio while thelimitation of the risk was variable and the riskwas expressed by a standard deviation NPVleading to effective portfolios, or, as the casemay be, to an efficient frontier (section 4.1).The task of setting possible impacts ofincreasing or decreasing disposable volumeof certain limited sources on the structure andthe effects of the portfolio dealt with in section 4.2is conceptually similar and also significant fromthe practical point of view. Further two tasksaim partly at the maximization of exceedingthe target value of the optimization criterion, inour case it is NPV (section 4.3), and partly atthe minimization of exceeding the capitalbudget as a key limitation of the development ofa project portfolio under risk (section 4.4).

4.1 Efficient FrontierBy means of the optimization tool OptQuest fivetasks of stochastic optimization were solved,with a criteria function formed by the mean NPVvalue, two source limitations (the capital budgetCZK 560 million and the number of workers notexceeding 240) and the gradually releasedportfolio risk limitations with regard to NPVmeasured by its standard deviation. The upperlimit of the standard deviation was graduallyincreased by CZK 1 million, from CZK 5 millionto CZK 9 million. During the time of the tasksolving in the length of two minutes theprogramme came up with 2,304 solutions, outof which 1,816 were feasible. The results of thesolution are summarized in Table 4.

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Tab. 3: A Model of an Investment Portfolio

Project E(NPV) σ(NPV) E(IN) σ(IN) Number(mil CZK) (mil CZK) (mil CZK) (mil CZK) of workers

Project 1 22.30 3.20 31.0 2.30 23

Project 2 5.16 1.62 27.6 1.70 12

Project 3 28.21 6.67 126.8 4.10 45

Project 4 26.32 4.35 96.5 6.80 37

Project 5 15.32 1.53 55.8 4.60 25

Project 6 10.47 1.80 36.8 3.00 23

Project 7 12.30 0.94 44.7 3.80 15

Project 8 23.01 2.75 67.5 4.30 26

Project 9 15.07 2.11 49.0 3.30 24

Project 10 29.47 5.32 85.8 5.50 38

Project 11 20.24 3.91 53.3 4.70 20

Project 12 12.35 3.12 42.3 3.10 17

Note: E(NPV) – mean value NPV for a given project,σ (NPV) – standard deviation NPV for a given project,E(IN) – mean value of investment costs of a given project,σ (IN) – standard deviation of investment costs of a given project. Source: own calculations

Tab. 4: The Results of Stochastic Optimization with Variable Limitations of a PortfolioRisk

Effective PortfolioE(NPV) σ(NPV) E(U) E(IN) P

portfolios structure

EP1 4–7, 9 99.7 4.9 3.85 336 144

EP2 4–10 132.0 5.8 4.46 436 188 35.9

EP3 1, 2, 4–10 159.5 6.8 5.57 495 223 27.5

EP4 1,2, 4–10, 12 171.8 7.5 6.31 537 240 17.6

EP5 1,2, 4, 5, 7–12 181.5 8.6 6.05 554 237 8.9

Notes: see Tables 1 and 3 Source: own calculations

∆E(NPV)––––––––σ∆(NPV)

As is obvious from Table 4, with gradualsoftening of the requirements concerning theportfolio risk measured by the standarddeviation σ(NPV), the mean value NPV ofportfolios increases and their mean utility valuealso grows. With the first two effective portfoliostheir risk limitation does not enable the fullexploitation of sources. The last two effectiveportfolios lead to almost full exploitation ofsources. The last effective portfolio EP5 leadsto the maximum mean NPV value. The

effective portfolio EP4 reaches a higher meanutility value with a lower risk. When choosingbetween these two effective portfolios it shouldbe, therefore, necessary to consider the growthof the mean NPV value when transferring fromportfolio EP4 to portfolio EP5 on one hand, andthe risk increase and the decrease of the meanutility value on the other hand. The division ofthe probability of the effective portfolio EP5 withthe maximum mean value of its NPV isillustrated by Figure 1.

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As is obvious from Figure 1, NPV of thisportfolio varies between CZK 146.2 million toCZK 214.1 million, and the probability that theNPV will reach the value from the interval fromCZK 170 million to CZK 190 million is approximately75%. The skewness -0.0273 demonstratesa slight deviation of this division towards thelower NPV values. The balancing of thisdivision by the most suitable of the theoreticaldivision according to Anderson-Darling test ledto a normal division with identical parameters ofits division as set by the simulation.

From Table 4 it is also obvious that byincreasing the risk of the effective portfolios thegrowth of the NPV mean value on a risk growthunit as measured by a standard NPV deviationdecreases (see column ∆E (NPV)/∆σ(NPV).This fact is also confirmed by the graphic imageof the effective portfolios in the form of theefficient frontier on Figure 2, when the slopes ofthe line segments connecting the individualeffective portfolios decline from the left to theright. See [32], [33] for more about this approach.

Fig. 1: Division of NPV Probability of Portfolio 5


Fig. 2: Efficient Frontier


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4.2 The Influence of the SourceLimit Increase on the PortfolioEffects

As is obvious from Table 4, the effectiveportfolio EP5 with the maximum mean NPVvalue leads to almost a full use of the limitedsources. It might, therefore, be useful to find outto what extent gaining additional sources mightincrease the effects of the portfolio optimization.

The chosen results of stochastic optimiza-tions with a somewhat increased capital budgetand the limit of workers (again, the mean NPVvalue was the optimization criterion) are stated

in the Table 5. The portfolio D represents theoptimum (efficient) portfolio with the givencapital budget (CZK 560 million) and with thelimit of the number of workers (240), and it isidentical with the efficient portfolio EP5 from theTable 4. Even a slight increase of the capitalbudget (by CZK 3 million) and of the number ofworkers (by 8) leads to the optimum portfolioE with a higher NPV and utility mean values.Another optimum portfolio F shows a higherincrease of the capital budget (by CZK 30 million)and the limit of workers (by 20).

Tab. 5: The Influence of the Increase of Source Volume on the Optimum Portfolios

Port- EP Portfolio structure

E σ Efolios (IN) (NPV) (NPV) (U)

D 554 237 1, 2, 4, 5, 7–12 181.6 8.6 0.327 0.77 6.05 1.09 2.55

E 563 248 1, 4–12 186.9 8.7 0.331 0.75 6.47 1.15 2.61

F 590 260 1, 2, 4–12 192.0 8.8 0.325 0.74 6.83 1.16 2.63

G 608 250 1, 3–5, 8–12 192.3 9.6 0.315 0.74 6.10 1.00 2.44

Notes: see Table 1 and Table 3 Source: own calculations

E(NPV)–––––––

E(IN)E(NPV)–––––––

P

E(U)–––––––E(IN100)

E(U)–––––––

P100

In case of difficulties in increasing the limitof the restricted sources it would probably bepossible to implement the portfolio E that, withregard to the portfolio D, leads to the increaseof the mean NPV value and the mean utilityvalue to CZK 100 million of investment costsand to only to a slight increase of risk anda slight decrease of the mean NPV value onone worker. A higher availability of limitedsources (portfolio F) leads only to a very smallincrease in the mean utility value on the spentCZK 100 million and on one hundred workers,by decrease of similar indicators related to NPVbut it also leads to an insignificant risk increase.The last portfolio G requires, on top of portfolioF, gaining CZK 18 million and at the same timereducing the necessary number of workers by10. Most indicators characterizing this portfolioin Table 5 shows deterioration. Depending onthe possibilities of the increase of the volume oflimited sources portfolios E or F are possiblythe best to be taken into account.

4.3 Maximization of the Probabilityof Exceeding the CriterionTarget Value

Three optimization tasks were gradually solvedwith target NPV values amounting CZK 175 million,CZK 180 million and CZK 185 million and withtwo source limitations (the capital budget andthe number of workers). The results of thesolution are shown in Table 6. As is obviousfrom this table, the optimum portfolios H, I andJ maximizing the probability of exceeding thetarget NPV values are identical and are formedby ten out of twelve projects, while projects 3 and 6 were excluded.

The decrease of the probability ofexceeding the target value (from 77.4 % withportfolio H to 34.2 % with portfolio J) is theobvious consequence of increasing the targetNPV value of the portfolio.

The first three tasks were solved withoutany limitations related to the mean utility valueof the portfolio. If we required this value to

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reach at least 6.2, then with the target NPVvalue amounting CZK 180 million there isa change of the optimum portfolio, portfolio K inthis case. The mean utility value of this portfolioreaches the value 6.31, but there was a consi-derable decrease of the mean NPV value of theportfolio by approx. CZK 10 million. Theprobability of exceeding the target NPV valueby this portfolio is considerably low (only13.2%) and so it is highly probable (86.8 %)that this value is not going to be reached.

4.4 Minimization of the Probability ofExceeding the Capital Budget

The individual portfolios may vary by theprobability of exceeding the capital budget andtherefore this criterion can represent one of thecriteria of the portfolio assessment. In our casewe therefore solved a few optimization tasks ofthe minimization of the probability of exceedingthe capital budget amounting CZK 560 millionwith a gradually growing requirement for themean NPV value of the portfolio. The results ofthese tasks solution are summarized in Table 7(only mutually different solutions are stated here).

Tab. 6: The Results of Maximization of the Probability of Exceeding the Target NPV Value

PortfolioTarget value PortfolioNPV (NPVc) structure

P(NPV≥NPVc) E(NPV) σ(NPV) E(U)

H 175 1, 2, 4, 5, 7–12 77.4 181.5 8.6 6.05

I 180 1, 2, 4, 5, 7–12 57.0 181.5 8.6 6.05

J 185 1 , 2, 4, 5, 7–12 34.2 181.5 8.6 6.05

K 180 1, 2, 4–10, 12 13.2 171.8 7.5 6.31


Tab. 7: The Results of Minimization of the Probability of Exceeding the Capital Budget

Portfolio Lower limit P (IN ≥560) E(IN) E(NPV) σ(NPV) E(U) Projects E(NPV) (%) not included in

portfolio

L 171 0.01 507 171.5 9.0 5.72 2, 3, 5

M 171 4.39 537 171.8 7.5 6.31 3, 11

N 175 0.80 526 176.4 8.5 5.69 2, 3, 6

P 177 25.94 532 177.0 9.2 5.35 2, 5, 6, 7

R 180 30.48 554 181.5 8.6 6.05 3, 6


The portfolio L leads to an absolutelynegligible exceeding of the investment costsbut it has a considerably high risk as measuredby a standard NPV deviation. If we required thisrisk not to increase CZK 8 million, we wouldgain portfolio M with approximately the sameNPV value but with a considerably higher meanutility value (6.31). The probability that byimplementing this portfolio the capital budgetwill be exceeded is less than 5%. Out of othersolutions there is portfolio N that leads to a very

small exceeding of the capital budget (0.80 %),its mean NPV value being approx. by CZK 5million higher than with portfolio M but therewas a considerable decrease in the mean utilityvalue (from value 6.31 to 5.69). With otherportfolios, P and R, and considering higherrequirements for the mean NPV value, there isa considerable increase of the probability ofexceeding the capital budget. Figure 3 showsthe investment costs division probability of thelow risk portfolio M.

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4.5 Summary of the Results of theOptimization of a Project Portfoliounder Risk

The results of the solution of the individualoptimization tasks now enable to select suitablesolutions and to assess them with regard to theselected set of criteria. If we somewhat simplifythe portfolio assessment and consider thecapital budget as the only limiting source, then,with regard to a certain difference in thedemandingness of the individual portfolios onthe funds, it might be more appropriate not towork with the absolute criteria in the form of themean utility value or the NPV mean value butwith the relative values in the form of the meanutility or NPV value as related to the meanvalue of the unit of the investment costs spent.As for other suitable criteria for the portfolioassessment we might also consider the riskwith regard to the NPV expressed by the

standard deviation, the probability of exceedingthe NPV target value and the probability ofexceeding the disposable volume of funds.

The following selected results of theoptimization tasks were the assessed portfolioswhose common thread was the portfolio structure.� Port1 corresponds to portfolios C, D, I, R

and EP5 and it does not contain projects 3and 6.

� Port2 corresponds to portfolios K, M, EP4and it does not contain projects 3 and 11.

� Port3 corresponds to portfolio E and it doesnot contain projects 2 and 3 and its specificnature consists in the fact that the capitalbudget increases by CZK 3 million, to CZK563 million, and the number of workersincreases by 8, to 248.The values of the portfolio criteria together

with the mean values of utility and NPV areshown in Table 8.

Fig. 3: The Investment Costs Division Probability of Portfolio M


Tab. 8: The Criteria Values of Selected Portfolios

Portfolio σ P(NPV≥180) P(IN≥560)E(U) E(NPV) E(IN)

(NPV) (%) (%)

Port1 1.09 0.327 8.6 57.0 30.5 6.05 181.5 554

Port2 1.18 0.320 7.5 13.2 4.4 6.31 171.8 537

Port3 1.15 0.331 8.3 78.4 47.3 6.48 186.9 563


E(U)–––––––E(IN100)

E(NPV)–––––––

E(IN)

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It may now be possible to carry out theselection of the most suitable portfolio either inthe non-formalized way or by applying themethods of the multi-criteria assessment.� In case of the non-formalized approach we

would first compare the individual portfoliosfrom the individual criteria point of view, e.g.Port3 is better than Port1 from the point ofview of the mean utility and NPV value, butthey are approximately at the same level ofrisk etc. The choice of the portfolio, as far asthe risk is concerned, would be influencedby the attitude of the decision maker.

� The formalized assessment of portfolioswould require setting the significance of theindividual criteria in the form of their weightsand applying some methods of the multi-criteria assessment (e.g. [10]). This approachis suitable especially in case of a greaternumber of the assessed portfolios anda larger set of assessment criteria, where thenon-formalized approach may fail. In ourtask it would also be necessary to respectthe strong interconnection of the two followingcriteria, namely the size of the mean NPVvalue of portfolios and the probability ofexceeding the NPV target value.

Conclusion

The development of the project portfolios in theform of an investment programme or a researchprogramme in the business practice is accom-panied by a lot of drawbacks. Here are themost significant ones:� the way of handling risk (the portfolio is

mostly created in the quasi conditions ofcertainty; i.e. on the basis of the only onepossible scenario of the future development);

� the projects are assessed and included intothe portfolio independently on one another;i.e. their mutual relations either of thedeterministic or stochastic character are notrespected;

� the multi-criteria character of the task isonly seldom respected in the optimizationof the portfolio.These and some other drawbacks of the

portfolio development, including for examplethe absence of the projects interlinked with thecompany strategic objectives, non-respectingthe various levels of risk of the individualprojects, the imbalance of the portfolio from the

point of view of the represented project types,the intuitiveness of the creation without applyingthe analytic tools based on quantitative dataare pointed out by for example [19], [21], [20],[27] and [26]. The political character of theprocess of the portfolio development can alsohave negative impacts on the companyeffectiveness. Instead of handling this issue asa rational process considering the sourcedemandingness of the individual projects, theircorporate benefits, risks and mutual dependenciesit is rather a process that is insufficientlytransparent, aimed at pushing local interests,compromising, demonstration of power(Sanwall [28] calls this process Decibel-Driven)and such like.

The above drawbacks can be eliminated, orat least reduced by the application of theoptimization models, in which we characterizedpartly the optimization of a project portfoliodevelopment under risk in terms of thedeterministic equivalents of this task by meansof some models of bivalent programming, andpartly the stochastic optimization based on theMonte Carlo simulation. Gaining suitable ITsupport may not be enough for practicalapplication of these tools, which undoubtedlyprovide invaluable information for creatinginvestment or research programmes, but it isalso necessary to create the needed personneland organisational prerequisites (includingproviding the necessary know-how) and toovercome the resistance towards change bymeans of suitable motivation and first of all withthe help of the company top management (forfurther detail see [14]). With regard to the factthat the decision making process concerningthe investment programmes or research pro-grammes belongs to the key decisions ofa strategic nature, the optimization of a projectportfolio development under risk may becomean important factor of the growth ofeffectiveness and prosperity in any company.

The article was worked out as one of theoutputs of the research project called“Competitiveness” (registration number VSEIP300040) solved at the University ofEconomics, Prague, Faculty of BusinessAdministration funded by institutional support ofthe long-term conceptual development atresearch institutions.

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prof. Ing. Jifií Fotr, CSc.University of Economics, Prague

Faculty of Business [email protected]

Ing. Lenka ·vecová, Ph.D.University of Economics, Prague

Faculty of Business [email protected]

doc. Dr. Ing. Miroslav Plevn˘University of West Bohemia in Pilsen

Faculty of [email protected]

doc. Ing. Emil Vacík, Ph.D.University of West Bohemia in Pilsen

Faculty of [email protected]

Doruãeno redakci: 29. 3. 2013Recenzováno: 22. 5. 2013, 30. 5. 2013Schváleno k publikování: 27. 9. 2013

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Abstract

MULTI-CRITERIA PROJECT PORTFOLIO OPTIMIZATION UNDER RISK ANDSPECIFIC LIMITATIONSJifií Fotr, Miroslav Plevn˘, Lenka ·vecová, Emil Vacík

The development of a portfolio of investment projects is a relatively underestimated economicpractice, which often leads to wrong investment decisions with a negative impact on the corporateperformance. This development is often done under certainty, which means with the only onepossible scenario. The multi-criteria nature of the task character is also rarely respected. Theevaluation of projects is usually done in isolation, without any connection to other projects orwithout taking into account dependencies between the projects.

This paper aims to specify the problem of optimization of development of a project portfoliounder risk (optimal allocation of scarce resources). The article offers two approaches to theoptimization. The first approach is based on deterministic equivalents with application of bivalentprogramming (multi-criteria and mono-criteria optimization). The second one uses stochasticoptimization based on the Monte Carlo simulation. The application of these model approaches cangreatly improve the quality of the project portfolio development under risk.

Key Words: Project portfolio development, simulation Monte Carlo, investment projects, risk.

JEL Classification: C15, C61, D81, L25, M21.

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multi-criteria project portfolio optimization under … multi-criteria... · multi-criteria utility...

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