decision-making process for project portfolio management

160 Int. J. Services Operations and Informatics, Vol. 6, Nos. 1/2, 2011

Copyright © 2011 Inderscience Enterprises Ltd.

Decision-making process for project portfolio management

Fateh Belaid CNRS/UMR 7218 LAVUE, ENSA PARIS, 3-15 Quai Panhard et Levassor, 75013 Paris, France Email: [email protected]

Abstract: The purpose of this paper is a quantitative study of the risk surrounding the main problem of investment decision making in project portfolio selection. The primary objective is to provide managers with a decision support tool allowing them to choose carefully their investment strategies and reduce risk of failure. The developed computer model consists of three major modules. The first module calculates projects cash-flow using a deterministic method. The second module calculates expected after tax net cash flows and estimates performance indicators for each realisation, thus yielding distribution of return for each project. The third module selects a set of projects (portfolio) using their covariance and semi-covariance matrix. In summary, we will define a selection model for portfolio management of investment projects that will not only take into account risk, but also the effect of the interdependence of projects. The model will be applied to a portfolio of projects in petroleum upstream.

Keywords: project management; decision making; portfolio optimisation; risk analysis; Monte Carlo simulation.

Reference to this paper should be made as follows: Belaid, F. (2011) ‘Decision-making process for project portfolio management’, Int. J. Services Operations and Informatics, Vol. 6, Nos. 1/2, pp.160–181.

Biographical notes: Fateh Belaid is currently a postdoctoral fellow at the Habitat Research Center, part of National Center for Scientific Research (CNRS), UMR 7218. He received his PhD in economics and management from the University of Littoral Côte d’Opale. His research focus is on decision theory, risk analysis, and portfolio project management.

1 Introduction

1.1 Context

In the capital-intensive industries, a firm will normally invest in a portfolio of investment projects rather than in a single project. The problem with investing all available funds in a single project is, of course, that an unfavourable outcome could have disastrous consequences for the company. In general, the funds are not sufficient to support all available investment opportunities. So the selection of portfolio projects is important.

Decision-making process for project portfolio management 161

The selection process objective is to choose a subset of projects that maximises profits (objective) of the company while respecting the budget restriction (constraints). However, in addition to all the above considerations, the company should follow a strategy to ensure better returns with minimal risk.

Premium evaluation criteria in investment are necessary in each step of the decision. However, the important question that needs to be answered in this research is the following: how shall we use the portfolio optimisation approach to make the strategic investment decision as easy as possible from a practical point of view?

Turner (1992) defines investment as follows: “An endeavour in which human, material and financial resources are organised in a novel way, to undertake a unique scope of work of given specification, within constraints of cost and time, so as to achieve unitary, beneficial change, through the delivery of quantified and qualitative objectives.”

This definition highlights the changes brought about by the nature of projects, the need to organise a variety of resources subject to significant constraints, and the central role given to defining the objectives of the project. Turner also suggests paying special attention to uncertainties inherent to the new organisation as a central element of effective project management. The competitiveness of a firm especially in the capital-intensive industries is highly dependent on a stream of successful and profitable projects. Economic analysis of the potential profitability of projects is thus an important task that demands efficient methods for evaluation and selection of projects.

For example, in the petroleum industry, the exploration and development of an oil field face many unknowns uncertainties associated with yields and costs throughout the life cycle of the project: capital expenditure (Capex), the operating costs (Opex), the production rate, the oil price (and gas), the rate of geological success and train of expenditure, particularly for sub-sea wells (the offshore projects). With all these uncertainties it is extraordinarily difficult to forecast profits and cash flow, for even the simplest prospects. The economic risk of petroleum projects is essentially linked to the economic environment. In particular, the evaluation of the profitability of investments is based primarily on scenarios of oil prices, so that the latter is the main determinant of income.

1.2 Motivations

Project portfolio management is a complex decision-making process involving the unrelenting pressures of time in cost. This decision-making process is affected by many critical factors such as the market conditions, raw materials availability, probability of technical success and government regulations. The problem of project selection was first developed in the field of engineering economy (DeGarmo and Canada, 1973). The problem has been given full attention in engineering management (Martino, 1995; Jafarizadeh and Khorshid, 2008; Wang et al., 2009).

The traditional approach considers corporate projects as being independent of each other. Yet, the relations and correlation between projects within the uncertain environment have been recognised as a major issue for corporations. Therefore, research in this field has recently shifted towards project portfolio management (Jonas, 2010; Laslo, 2010). The present paper focuses on the new project management approach.

For the petroleum industry, the development of the collection and analysis of seismic data significantly reduced the risk of undiscovered oil. The geology and geophysics resultant (G&G) have revolutionised oil and gas exploration. Decision analysis has traditionally been applied to information derived from G&G for ranking projects hole by

162 F. Belaid

hole, determining on an individual basis whether they should be explored and developed. This ‘hole-istic’ approach is being challenged by a holistic one that takes into account the entire portfolio of potential projects as well as current holdings. For Ball and Savage (1999), this portfolio analysis starts with representations of the local uncertainties of the individual projects provided by geology and geophysics. It then takes into account global uncertainties by adding two additional ‘G’s: geo-economics and geopolitics, thereby reducing risks associated with price fluctuations and political events in addition to those addressed by traditional G&G analysis.

This paper presents comments and some background on the use of risk analysis methods in the selection of investment projects. We apply variant models of Modern Portfolio Theory to determine the efficient project portfolio that maximise the expected profit while simultaneously minimising risk. Our approach is generic while particular attention is given to applications developed for exploration and production projects selection.

Portfolio optimisation is a methodology from finance theory for determining the investment programme that gives the maximum expected value for a given level of risk or the minimum level of risk for a given expected value. In his seminal paper published in 1952 in the Journal of Finance, Nobel laureate Harry Markowitz laid down the basis for modern portfolio theory (Markowitz, 1952a). Markowitz focused the investment profession’s attention on mean-variance efficient portfolios and opened the area of modern investment theory. The methodology of the model presented in this paper is derived from two sources: decision analysis and portfolio theory.

1.3 Contributions

The main contributions of this paper are as follows:

1 The model proposed illustrates how modern portfolio theory provides management with a superior setting for allocating capital by illuminating risk at the portfolio level. This method, contrary to the classical selection methods, analyses the projects by considering the various aspects of the risk and the various correlations existing between the projects.

2 The scenario method is proposed for the assessment of crude oil price volatility.

3 The proposed method is flexible in terms of input data and output results and simple to implement.

The rest of the paper is organised as follows: Section 2 presents the state of the art dealing with project selection and portfolio management. Section 3 details the proposed model. Section 4 presents the details of the model with an illustrative example. Section 5 deals with the results of the proposed project selection example. Section 6 concludes the paper.

2 Literature review

The problem of project selection was first introduced in the field of engineering economy. According to Bussey (1978), the early practitioners had to consider the problem of selecting the economic choice between executable alternative solutions. This


approach considers corporate projects as being independent of each other. Jafarizadeh and Khorshid (2008) stated that the field of industrial project selection was born by the introduction of the question: “What is the best for the firm, overall?”. This field is concerned with the act of project selection in the context of ‘firm’. Now, the problem has been given full attention in engineering management. Featured as a process of strategic significance, Dey (2006), Corvellec and Macheridis (2010) and Mantel et al. (2008) defined project selection as “the process of evaluating individual projects or groups of projects and then choosing to implement a set of them so that the objectives of the parent organisation are achieved”.

Mohanty (1992) classified the criteria influencing project selection into two categories: intrinsic and extrinsic. Kaplan and Norton (2001) confirmed the importance of project selection by stating that a critical component for linking strategy to short-term actions is opting for new initiatives. Several methods have been proposed to help organisations make good decisions for project selection problem (Anadalingam and Olsson, 1989; Wang et al., 2009).

In the field of the petroleum industry, decision analysis was first applied to E&P projects by Allais (1956) with his study of the economic feasibility of exploration in the south of Algeria. There were several further attempts to apply decision analysis to petroleum upstream by Grayson (1960), Krumbein and Graybill (1965) and Drew (1967). These concepts have been popularised by Cozzolino (1977), Harbaugh and McDonald (1984), Harris (1984, 1990), Newendorp and Schuyler (2000), etc. Cozzolino (1977) used an exponential utility function in the determination of future cash flows of an oil exploration project to express the certainty equivalent, which is equal to the expected value less compensation risk, called risk premium. Another important contribution was made by Walls and Dyer (1996). Walls incorporated the Multi-Attribute Utility Theory Approach in the selection process for upstream projects. This approach provides a rich insight into the effects of the integration objectives of the oil companies and the analysis of risk in investment choices. Walls and Dyer (1996) used this approach to study changes in the propensity of risk depending on the size of companies in the oil industry.

The evolution of quantitative analysis of the portfolio began in the 1950s, with the revolutionary work of Markowitz (1952a), who popularised the idea that increasing returns implies increased risk. Markowitz developed the mathematical basis and consequences of this analysis in his thesis in 1954. Sharpe (1964) extended and developed the work of Markowitz, with his model of asset pricing (CAPM: Capital Asset Pricing Model), while Modigliani and Merton (1958) presented another important contribution to the theory of values assessment. In the early 1970s, Black and Scholes (1972) and Merton (1973) determined the principle of rational evaluation of stock options. Since then, several studies have been conducted in this area.

Hertz (1968) discussed the application of the portfolio theory model to risky industrial projects with respect to the way it is used in the financial market. In 1983, Ball and Savage presented an application of decision analysis to the management of risk and return in petroleum exploration. Since 1990, these two authors have collaborated on a set of models to meet the needs of certain companies. This has helped to refine the method and to facilitate its application to E&P projects (exploration and production). In addition, in 1997, the Earth Observatory Lamont-Doherty of Columbia University has founded a consortium of oil companies to share knowledge in E&P project portfolio analysis based on Ball and Savage (1999) models, Holistic vs. Hole-istic E&P Strategies. This model remains one of the most popular in the area of E&P project management. Since then, several studies have been conducted in this direction (e.g. Walls, 2004; Erdogan et al.,

164 F. Belaid

2001). The original idea is that a portfolio may be worth more or less than the sum of its component projects, and there is not a better portfolio, but a family of optimal portfolios that allow for achieving a balance between risk and return. Jafarizadeh and Khorshid (2008) proposed a method of project selection based on capital asset pricing theories in framework of ‘mean semi-deviation behaviour’.

The research work presented in this paper shows the importance of the use of the Modern Portfolio Theory in project portfolio management. The proposed model includes a methodology to improve the quality and efficiency of the decision-making process of project portfolio management.

3 Proposed model

The process we use to optimise the exploration or production projects portfolio is an integrated economic model. Implementation of this process involves three basic steps focused on three main stages:

1 Individual evaluation of each project by using the deterministic method to evaluate the cash flows of oil exploration projects or production projects.

2 Applying Monte Carlo simulations for the assessment of projects economic risk.

3 Building a portfolio optimisation model for the selection of the optimal portfolios.

The construction and operation of the model are summarised in Figure 1.

Figure 1 Model flow of optimisation process

Project Evaluation

Monte Carlo Simulation

Optimisation Process

Evaluation of every individual project using deterministic approach for the determination of the main decision criteria: NPV.

A Monte Carlo simulation of the joint economic outcomes of the projects is run for risk assessment, based on the local uncertainties (geo-science), and the global uncertainties (geo-economics and geo-politics). The statistical dependence between projects must be preserved.

An optimisation of linear/quadratic program for all the projects to find the optimal capital allocation to each project. In such ways that return is maximised for a certain level of risk.

The aim is to obtain an optimal rate of investment for each project to maximise the total return of the portfolio taking into account the various risks and the constraints of the company. To do this, we compare two methods: a model inspired by Modern Portfolio Theory method with the variance as a risk measure and a model inspired by Modern Portfolio Theory method with the semi-variance as risk measure.


We apply this process to a real problem of project evaluation and selection. The selected projects for our problem are a group of 14 real petroleum projects. The decision problem is: which ones of the mutually exclusive projects are the best for the company considering their risk and return?

3.1 Deterministic evaluation

The first step in evaluating a project is to create a scenario of the baseline situation and calculate the indices of efficiency. The main indices used to estimate the investment projects are described in Table 1. Table 1 Main indices of project profitability

Index Meaning of index Criterion for positive estimation of the index Calculation algorithm

Net present value (NPV)

Excess of the total cash income over the total expenditures for the given project with regard for discounting

NPV > 0 ( ) ( )( )1 1

T

tt

R t C TNPV K

E=

−= − +

+∑

Internal rate of return (IRR)

Positive number IRR such that for the discount rate E=IRR the net discount income of the project vanishes, for E>IRR is negative, for E<IRR is positive

IRR > E ( ) ( )( )1

01

T

tt

R t C TK

IRR=

−− + =

+∑

IRR is the positive root of equation

Payback period (PB)

Time from the beginning of project realization to the instant when the current discounted net income becomes nonnegative and remains so

PB ≤ T, T is the time of diversion of investment resources acceptable for investor

Minimum time T beginning from which observed in the equality

( ) ( )( ) ( )1 11 1

T Tt

t tt t

R t C T KE E= =

−≥

+ +∑ ∑

Profit index (PI)

Ratio of the sum of discounted cash inflows to the sum of discounted cash runoffs, shows the relative increment of NPV

PI > 1 ( ) ( )( )1 1

Ttt

R t C TE

PIK

=

⎛ ⎞−⎜ ⎟⎜ ⎟+⎝ ⎠=∑

Source: Karibskii et al. (2003)

In our case we use Net Present Value (NPV), the most popular index used in discount cash flow project evaluation. This assumes that the values of input parameters are known, namely:

• the nature of oil in place

• the production of year t, noted Pit

• oil prices for year t, noted Pricet

• the capital expenditure, noted Capexi

• the operating cost, noted Opexi

• the tax of year t, noted Taxest.

166 F. Belaid

In our case we used the following formula:

( )( ) ( )

1

1

nit i it

i t

P Price Capex Opex TaxesNPV

r p r p=

× − − −=

+ + + ×⎡ ⎤⎣ ⎦

∑

with r = the discount rate and p = inflation rate. A positive Net Present Value means that the investment increases the company’s

value. A negative Net Present Value means that the investment reduces the value of the company, and the productivity is lower than the cost of capital. A positive NPV will lead to the acceptance of the project and a negative NPV will cause its dismissal.

To calculate the NPV, we need to know the cost of the capital. The determination of the discount rate is a critical element of economic calculation. The estimation of a suitable discount rate is often the most difficult and uncertain part of a DCF (Discount Cash Flow). This is only worsened by the fact that the final result is very sensitive to the choice of discount rate – a small change in the discount rate causes a large change in the value.

The concept behind the cost of capital is simple: the compensation of providers of capital for (a) the time value of money and (b) the risk that they agree on the expected cash flows (not materialised as expected). Obtaining an estimation of a necessary compensation is not so simple. This is because to estimate the cost of capital, we have to estimate the risk of the project and necessary performance to offset this risk.

3.2 Stochastic evaluation

The simulation allows analysts to describe the risk and the uncertainty of variables that influence the profitability of the project with the help of probability distributions. Examples of uncertain variables in petroleum industry are the reserves, the drilling costs, the crude oil price, etc.

The first objective of the use of simulation in the evaluation of investment project is to determine the distribution of the NPV from the variables that affect project performance, resulting in its average or the Expected Net Present Value.

The current situation in the oil industry is that we do not know the exact values of the parameters. But if we can describe the interval and the possible values of each random variable, we can use simulation to generate the distribution of the resultant NPV. The idea is that, from a simple equation, we can use the model of the project as an equation for the NPV.

We performed a stochastic assessment for each project using the Monte Carlo simulation. First, we determine the distributions of input parameters (Production, Capex and Opex) and secondly we estimate the distribution of the output parameter which is the NPV for our case, from the Monte Carlo simulation. From this we calculate the NPV and average variance. Finally, we retrieve the data from the simulation to calculate the correlation between projects, then the matrices of variance-covariance and semi-covariance which can be used in an optimisation model.

We can summarise the process of Monte Carlo simulation conducted in three main steps, as follows:


Step a: Creation of a distribution for each input parameter: first we must identify the main risk factors, which are here production, opex, capex and assess their distributions using the historical values and the expert judgements. In our example distributions allocated to the main variables are:

1 Production: log-normal distribution (see Figure 2).

2 Capex (investment): triangular distribution (see Figure 3).

3 Opex (operating costs): triangular distribution (see Figure 4).

Recall that these distributions are frequently used in the petroleum industry, e.g. (Rose, 1987; Orman and Duggan, 1998; Rodriguez and Oliveira, 2005).

Step b: Generation of a Monte Carlo simulation with 5000 trials.

Step c: Recording the results of the simulation (the distribution of the expected NPV, the average of the NPV, its variance). Figure 5 shows the results of one particular project’s Monte Carlo simulation.

Figure 2 Distribution of production

Figure 3 Distribution of investment costs

168 F. Belaid

Figure 4 Distribution of operating costs

Figure 5 Probability distribution of project 1 and its statistics

In the current context (the high volatility of crude prices), it is difficult to detect a long-term relationship that can describe the evolution of prices. Consequently, the forecasting model will fail, knowing that petroleum projects have a long duration (5–30 years). To overcome this problem, in calculating the NPV, the ideal would be to imagine a few price scenarios (three) taking into account all factors of the current economic climate, and possible future changes (continuously increasing the global demand, possible depletion of reserves, discovery of new deposits, possible arrival of a new energy, etc.).


For our problem of the project portfolio evaluation, we use three different scenarios for the crude price: a low price, an average price and a high price scenario. From these scenarios, we build four models:

Model 1: This model corresponds to the low price scenario. This scenario assumes a soft landing. The growth of world oil demand is slowing sharply because of the transition to alternative energy, and oil prices fall to an average price of $25/barrel.

Model 2: This model corresponds to the scenario of an average price. This scenario is based on a continuity of supply of crude oil. Global demand remains strong but begins to slow down. The price of a barrel of crude sells at around $100/barrel.

Model 3: This model corresponds to the scenario of a high price. This crisis scenario provides that the supply will be disrupted by terrorist attacks or political unrest, and it will no longer be able to meet demand. Crude price increases to $200 per barrel.

Model 4: a mixed model where we assign a subjective probability of occurrence of the three price scenarios:

• the price $25 per barrel with a probability of 0.2

• the price $100 per barrel with a probability of 0.4

• the price $200 per barrel with a probability of 0.4.

In the latter scenario, we calculate, for each project, the NPV for the three price scenarios. Then, we calculate the expectation of NPV taking into account the probabilities of occurrence of the three scenarios.

4 Optimisation and selection of optimal portfolio

Since the seminal work of Markowitz (1952a, 1952b), mathematical analysis on portfolio management has grown considerably. Variance has become the most popular mathematical definition for the risk of portfolio selection. Markowitz shows how rational investors can construct optimal portfolios under conditions of uncertainty. The mean and variance of a portfolio’s return represent the benefit and risk associated with the investment. Several researchers have developed a variety of models to handle risk using variance as risk measure, e.g. Chopra and Ziemba (1998), Chow and Denning (1994) and Hlouskova (2000).

Variance is a useful measure of risk. In calculating variance, positive and negative deviations from the mean are equally weighted. In fact, decision-makers are often more pre-occupied with downside risk – the risk of failure. A solution to this problem is to determine the efficient set of portfolios by using another risk measure. Semi-variance overcomes this problem by measuring the probability and distribution below the mean return. Several models have been developed by using semi-variance as risk measure, e.g. Homaifar and Graddy (1990), Huang (2008), Grootveld and Hallerbach (1999), Markowitz et al. (1993), etc. In the mean downside risk investment models the variance is replaced by a downside risk measure, then only outcomes below a certain point contribute to risk.

170 F. Belaid

For our problem, a portfolio optimisation was conducted for our group of 14 petroleum upstream projects under risk consideration, resulting in an efficient frontier. In this case, risk was measured by the variance as calculated across Monte Carlo simulation. In this model the objective function is a linear combination of net present value against risk (variance).

4.1 Notations

For indices, we use, i and j to denote the different E&P projects and t for the years.

We used the following two sets:

N: the total number of projects (14 in our application)

E: the set of specific projects (5 in our application)

ENPVi: the expected return of project i

σij = coefficients of the covariance matrix defined for the NPV of projects i and j (given by the Monte Carlo simulation)

Ωij = coefficients of the semi-covariance matrix for the NPV of projects i and j (given by the Monte Carlo simulation)

Ri = reserves of project i for year t

Pit = production of project i

Ii = investment of project i

Pmin t = minimum targeted production for year t

Rmin = minimum reserves required from the selected portfolio (500 million barrels in our example)

Imax = total capital available to investment (5000 $MM in our example)

α = capital enrichment for the selected portfolio

ρmin = the desired level of return for the selected portfolio (900 $MM in our example)

λ = coefficient reflecting the risk aversion factor of the investor (0 ≤ λ ≤ 1)

β = the minimal fraction of the total investment allocated to exploration product (20%)

4.2 Optimisation model with variance as risk measure

The decision variables are simply Xi, i = 1, 2,…, n; the fractions of the project i invested in the portfolio. We suppose that we can participate in all projects at any level (thus 0 ≤ Xi ≤ 1). The model of optimisation with variance as risk measure is written below:

Model: Max (NPV–VAR)

( )11 1 1

Max 1N N N

i i i j iji i j

Z X ENPV X Xλ λ σ= = =

= − ⋅ −∑ ∑∑


Under the constraints: 1... ; 2013...2017i N t∀ = ∀ =

max1

min1

min1

1, 1

min1

0 1

I

0

iN

i iiN

i iiN

i ii

N N

i i i ii i E i

N

i it ti

X

X I

X R R

X ENPV

X I X I

X P P

ρ

β

=

=

=

= ∈ =

=

≤ ≤

⋅ ≤

⋅ ≥

⋅ ≥

⎛ ⎞ ⎛ ⎞⋅ − ⋅ ≥⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

⋅ ≥

∑

∑

∑

∑ ∑

∑

It’s a quadratic model maximising the objective function (return – risk). For each value of λ, we determine the value Xi (fraction of each project invested in the portfolio) and the corresponding portfolio return and risk. The latter are elements of the efficient frontier. We get the efficient frontier point by point by solving this problem for different values of λ. The values of λ are shown in Table 2. Table 2 Risk aversion coefficients used in the optimisation process

λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8 λ9 λ10 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

λ11 λ12 λ13 λ14 λ15 λ16 λ17 λ18 λ19 λ20 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1

4.3 Optimisation model with semi-variance as risk measure

Variance penalises both extreme gains and extreme losses. In contrast, semi-variance stimulates the selection of projects with a probability of returns below a critical value like the expected return. Therefore, we propose a new portfolio optimisation model with semi-variance as risk measure. Each portfolio in the frontier represents a set of projects satisfying minimum levels of production, tolerable operational and maintenance costs, precedence relations between projects, and a limited budget.

Likewise, according to prospect theory (Tversky and Kahneman, 1991), investors see a loss as riskier than a gain of the same amount, which contradicts the use of variance when extreme potential gains and losses are equivalent (Choobineh and Branting, 1986). Semi-variance overcomes this problem by measuring the probability and dispersion below the mean value. In accordance, the optimisation framework is now modified to minimise semi-variance (instead of variance) for a given expected NPV.

172 F. Belaid

The model of optimisation with semi-variance as risk measure is written below:

Model: Max (NPV–SVAR)

( )11 1 1

Max 1N N N

i i i j iji i j

Z X ENPV X Xλ λ= = =

= − ⋅ − Ω∑ ∑∑

Under the constraints: 1... ; 2013...2017i N t∀ = ∀ =

max1

min1

min1

1, 1

min1

0 1

I

0

iN

i iiN

i iiN

i ii

N N

i i i ii i E i

N

i it ti

X

X I

X R R

X ENPV

X I X I

X P P

ρ

β

=

=

=

= ∈ =

=

≤ ≤

⋅ ≤

⋅ ≥

⋅ ≥

⎛ ⎞ ⎛ ⎞⋅ − ⋅ ≥⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

⋅ ≥

∑

∑

∑

∑ ∑

∑

5 Computational results and analysis

Economic evaluation models were created for each project in Excel file models and evaluated under concession contracts. For simulation, we use the latest version of Crystal Ball software version 7.3 (Charnes, 2007). The resolution of the different quadratic optimisation models under Markowitz’s method is obtained using GAMS software (Brook et al., 1992).

As in Markowitz et al. (1993), we compute the portfolio’s semi-variance as follows:

( )2

1

Ni i

i i i it

NPV ENPVSV S NPV ENPV

N=

−= <∑

The semi-covariance between projects is defined as follows:

( ) ( )1 1

min ,0 min ,0N N i i j jij

i j

NPV ENPV NPV ENPVS

Nσ

= =

⎡ ⎤− ⋅ −⎣ ⎦= ∑∑

We only present the results of semi-variance model.

Scenario 1 (oil price of $25/barrel): By using the semi-variance as a risk measure, we get 8 optimal portfolios. The portfolio P1 is the riskiest with a semi-variance of 1950.16 and a return of $1 336.04 million. P8 is the safest portfolio with a semi-variance of 908.11, a return of $1000 million, and an investment of $3874.90 million. The efficient frontier is presented in Figure 6. We notice that


this frontier is different from the one obtained with the use of the variance as the measure of the risk. For a similar return we have a lower risk. It is due to the fact that the semi-variance considers returns below the mean as risk, contrary to the variance which does not distinguish between the returns below and above the mean.

Scenario 2 (oil price of $100/barrel): The application of the method of Markowitz with the semi-variance as risk measure provides 19 optimal portfolios for this scenario. P1 is the riskiest portfolio with a semi-variance of 29,136, an output of $13,326.95 million and an investment of $5000 million, while P19 is the least risky portfolio with a semi-variance of 1938.45 and an expected NPV of $3913.17 million and one investment of $1938.45 million. The efficient frontier is presented in Figure 7.

Scenario 3 (oil price of $200 /barrel): The use of the semi-variance as the measure of the risk on this scenario gives us 19 optimal portfolios with compositions close to those obtained with the use of the variance. P1 is the riskiest portfolio with a semi-variance of 97,956.37, a $29,489.20 million dollar return and a $5000 million dollar investment. P19 is the safest portfolio, with a semi-variance of 4927.53, a NPV of $8721.27 million, and an investment of $1947.04 million. The efficient frontier is presented in Figure 8.

Scenario 4 (average NPV): This average scenario, gives us 19 optimal portfolios. Their compositions are different from those obtained with the variance as risk measure. On the other hand, the returns are close. P1 is the riskiest portfolio with a semi-variance of 49,882, a NPV of $17,367 million, and an investment of $5000 million (it is the same portfolio obtained with the classic method). P19 is the least risky portfolio with a semi-variance of 2484, a NPV of $5116 million, and an investment of $1952 million. This last portfolio reduces the risk by 95% and the investment ($3047 million) by 60% with a loss of 70% return ($12,250 million). The efficient frontier with the average NPV is presented in Figure 9. The results are presented in Table 3.

Figure 6 Efficient frontier for a price of $25/barrel

174 F. Belaid



Figure 9 Efficient frontier for the average price


Table 3 Scenario 4 (average NPV)

Port

folio

P1

P2

P3

P4

P5

P6

P7

P8

P9

P1

0 λ

(λ1)

(λ2)

(λ

3) (λ

4) (λ

5) (λ

6)

(λ7)

(λ

8)

(λ9)

(λ10

) EN

PV

17,3

67.5

2 17

,354

.58

17,2

27.8

1 17

,107

.88

17,0

42.5

9 17

,003

.71

16,8

52.4

8 16

,696

.74

16,5

75.1

2 16

,401

.37

SV

49,8

82.6

8 49

,159

.39

44,5

19.5

6 41

,114

.29

39,7

10.5

8 39

,033

.49

36,8

83.0

1 34

,953

.47

33,6

39.2

4 32

,011

.03

Inve

st.

5000

50

00

5000

50

00

4997

.72

5000

50

00

5000

49

98.7

75

5000

EX

P1

1 1

1 1

1 1

1 1

1 1

EXP2

0.

48

0.51

0.

56

0.67

0.

76

0.84

0.

82

0.79

0.

77

0.76

EX

P3

0 0

0 0

0 0

0.07

0.

17

0.24

0.

3 EX

P4

1 0.

89

0.84

0.

92

0.98

1

1 0.

96

0.93

0.

91

EXP5

1

1 1

1 1

1 0.

95

0.89

0.

84

0.81

EX

P6

1 1

1 1

1 1

1 1

1 1

EXP7

0

0.25

0.

84

1 1

1 1

1 1

1 EX

P8

1 1

1 1

1 1

1 1

1 1

EXP9

1

1 1

1 1

1 1

1 1

1 EX

P10

0 0

0 0

0 0

0.08

0.

17

0.23

0.

29

EXP1

1 1

1 1

1 1

1 1

1 1

0.97

EX

P12

1 1

1 1

1 1

1 1

1 1

EXP1

3 1

1 1

0.95

0.

9 0.

89

0.83

0.

78

0.74

0.

71

EXP1

4 1

1 0.

87

0.74

0.

67

0.62

0.

57

0.52

0.

49

0.46

176 F. Belaid

Table 3 Scenario 4 (average NPV) (continued)

Port

folio

P1

1 P1

2 P1

3 P1

4 P1

5 P1

6 P1

7 P1

8 P1

9 λ

(λ11

) (λ

12)

(λ13

) (λ

14)

(λ15

) (λ

16)

(λ17

) (λ

18)

(λ19

. λ20

) EN

PV

15,1

73.8

6 14

,514

.57

12,1

05.1

8,

340.

04

5721

.47

5498

.01

5361

.83

5180

.26

5116

.71

SV

23,0

09.3

4 19

,842

.74

13,2

71.2

60

60.4

4 28

02.7

5 26

11.1

7 25

36.6

24

87.6

24

84.0

7 In

vest

. 50

00

5000

43

29.1

4 29

63.7

7 20

20.5

13

1986

.934

19

67.7

05

1963

.717

19

52.7

49

EXP1

1

0.92

0.

66

0.43

0.

29

0.25

0.

22

0.18

0.

17

EXP2

0.

72

0.71

0.

6 0.

38

0.26

0.

24

0.24

0.

23

0.22

EX

P3

0.48

0.

58

0.56

0.

36

0.24

0.

25

0.25

0.

25

0.26

EX

P4

0.84

0.

81

0.67

0.

43

0.29

0.

31

0.34

0.

37

0.38

EX

P5

0.7

0.65

0.

52

0.33

0.

22

0.21

0.

2 0.

2 0.

19

EXP6

1

1 0.

88

0.57

0.

38

0.34

0.

31

0.27

0.

26

EXP7

1

1 1

1 0.

74

0.73

0.

73

0.72

0.

72

EXP8

1

1 1

1 1

1 1

1 1

EXP9

1

1 1

0.97

0.

64

0.54

0.

46

0.36

0.

32

EXP1

0 0.

45

0.53

0.

51

0.32

0.

21

0.22

0.

22

0.23

0.

23

EXP1

1 0.

62

0.44

0.

26

0.17

0.

11

0.08

0.

05

0.02

0

EXP1

2 1

1 0.

89

0.58

0.

39

0.33

0.

3 0.

25

0.23

EX

P13

0.62

0.

58

0.47

0.

3 0.

2 0.

2 0.

19

0.19

0.

19

EXP1

4 0.

38

0.34

0.

26

0.17

0.

11

0.11

0.

11

0.12

0.

12


5.1 Comparison of both resultant efficient frontiers of the maximisation of the variance and the semi-variance

To illustrate the impact of variance and the semi-variance as risk measure, we draw the efficient frontiers of both measures on the same graph for the various scenarios of price. The resulting efficient frontiers are plotted in Figures 10, 11, 12 and 13. We notice that the efficient frontiers of the models with the semi-variance as the measure of risk are different from the frontiers obtained using variance as risk measure. We note that for the same level of return, the semi-variance corresponds to a lower risk. However, with the diminution of risk, the difference or the dispersal between both curves becomes smaller.

Figure 10 Scenario 1 – oil price of $25/barrel


178 F. Belaid


Figure 13 Scenario 4 – average oil price

6 Conclusions

Analysis of obtained solutions has shown that the given optimisation model based on the modern portfolio theory method is robust and flexible in terms of input data and output results. However, the model allows the decision-makers to specify the best rate of participation in every project. Its modular architecture allows further inclusion of complementary constraints and utilises different measures (than considered) of risk and return as well as different input data formats. This method, contrary to the classical selection methods, analyses the projects by considering the various aspects of the risk and the various correlations between the projects, such as places, prices, profiles, politics, and procedures. This allows to better take into account the risk of the projects, and limiting the investment failures. More specifically the method allows determining: • the allocation of capital as well as the distribution of the production and the returns

for each project

• the adequate strategies of investment, which answer the expected objectives, and respect the budgetary constraints of the company.


The use of variance as risk measure in the optimisation process penalises projects for upward as well as downward potential. We can overcome the problem by using semi-variance in the definition of the optimisation problem. Semi-variance concentrates only on the reduction of losses, without taking into account risks of the extreme potential gains. This approach allows the portfolio managers to define the risk in an adequate way according to the objectives and the constraints which concern their portfolio. This study demonstrates that an explicit analysis of uncertainties and interdependencies in evaluating the risks of E&P projects improves the quality of investment decision-making.

This paper is based on static evaluation of the risk of crude oil price and on a limited number of case studies, e.g. Rodriguez and Oliveira (2005); Jafarizadeh and Khorshid (2008). Therefore, empirical results cannot be generalised. Future researches to develop a dynamic model for the optimal portfolio selection incorporating the crude price uncertainty in a dynamic way are needed. Finally, we believe that the model proposed does not provide a certain and unique answer for the problem of project portfolio management but gives insights into what makes a desirable project portfolio for a company rather than an undesirable one.

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