Valuation of investment projects in the context of sustainable development

Real option approach

Krzysztof S. Targiel Department of Operations Research

University of Economics in Katowice Katowice, Poland

krzysztof.targiel@ue.katowice.pl

Abstract Traditionally, the evaluation of development projects is made on the basis of financial criteria. In the context of sustainable development, the social and environmental aspects must also be considered. Contemporary project valuation takes into account not only the static planned cash flows but also opportunities that may appear, known as real options. This paper describes a method for the valuation of development projects, taking into account the above aspects of valuation.

Keywords- sustainable development; project valuation; real options

I. INTRODUCTION Traditional project evaluation is based on the discounted

cash flow method (DCF). The main measure of effectiveness is the Net Present Value (NPV). When this value is positive, the project is approved, when negative, it is rejected, but this approach sometimes leads to the abandonment of profitable projects. The reason for this is that the DCF method does not take into account the role of managerial flexibility. The Project Manager has the right to take action as appropriate. This situation is called a real option. Using Real Option Valuation (ROV), we can provide a quantitative measurement for this situation.

In ROV, methods from the financial market, such as the Black-Scholes model [2] or the Cox-Ross-Rubinstein model (CRR)[7] were used first. An approach based on Monte Carlo simulations [3] has also been used. The CRR model is based on the binomial tree. This approach was also adopted in Guthries work [9], on which this study is based.

The traditional approach in the valuation of real options is based on one factor called the state variable. There have been attempts to take into account many state variables. The first attempt, on the basis of financial options, was made by Boyle [4], who took into account two assets. Mun [12] described the commercial solution as having such possibilities. Guthrie [9] also described problems for which it was necessary to take into consideration a number of variables.

In these solutions, different criteria were brought to a common financial denominator. The proposed solution considers various aspects such as multi-criteria, which is especially important in the context of sustainable development.

II. PROBLEM FORMULATIONProject management produces an environment in which

many decisions are made. Many project management methodologies, such as PRINCE2 [14], for example, recommend dividing the project into stages. This raises the problem of decision making, the choice of the start of the next steps.

According to the PRINCE2 methodology, each project must have at least two stages of initialization and proper implementation. We will consider a project consisting of these two phases, each of which occupies a period of time. After initializing the project we have to decide when to begin implementation. We can delay the execution of one period. This is a classic example of the defer options considered by Ingersoll and Ross [10].

If steps are planned in advance, the situation is static; the decision maker is not able to react to changes in the environment and in the same project. If extending the duration of the project is allowed and the decision maker is allowed to freely decide about the start times of stages, a completely new situation is raised, as presented in Fig. 1. The decision maker may start the project (decision A), then move from the current state (initialization of project) to the last state (end of project). The decision maker may wait (decision W), but then the project will remain in its starting state.

Figure 1. Decision tree

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The results of the project, in terms of its value, depend on certain factors, which may be:

Economic factors (commodity prices, exchange rates)

Social factors

Environmental factors

according to the principles of sustainable development. If we consider more than one factor that leads to usability and design considerations in many areas, the problem is converted from a simple valuation to a multi-criteria evaluation problem.

This approach is discussed in the literature. Bednyagin and Gnansounou [1] consider the spillover benefits of the R & D program on thermonuclear fusion technology. The socio-economic benefits are higher than the expected costs. The framework for project-level decisions, leading to more sustainable management and development, is proposed in [6]. The ecological, economic and social sources of landscape valuation are discussed in [15]. Heidkamps paper [20] proposes a theoretical framework for the integration of economic aspects and environmental aspects into the decision making process for sustainable development strategies. The paper [21] incorporates sustainable development criteria into decision making processes within the industry. Renewable energy investment projects as an integral part of sustainable economic development are discussed in [5] with evaluating real options embedded in these projects.

The decision maker makes its decisions based on the changes in factors. These factors vary stochastically according to a certain random process. The idea behind the CRR method is to cover a possible future state variable binomial tree as shown in Fig. 2. This is a role-specific scenario of possible changes in the value of state variables. At each stage, we consider only the possibility of an increase or decrease in value. With this procedure the decision making process is simplified.

This issue is discussed in the literature by Guthrie [9]. Guthrie considered one such factor, and this article extends these considerations and binds the results of the project with two factors.

Figure 2. The binomial tree covering the stochastic process

III. METHOD OF EVALUATIONWe assume that each state variable in one period may

increase u-times and fall d-times. This assumption leads to the tree of possible state variable values, consisting of nodes marked with indices (i, n) where i means number of falls and nmeans period number.

A state variable and cash flow is connected to each node. We denote it as (in each node (i,n)) :

Xk(i, n ) k-th state variable in period n

Ym(i, j, n) - cash flow in period n (where m is state of project)

We assume that we have N periods, the present value of each state variable denoted by Xk(0,0) and also u and d. The value of u can be obtained from historical data by the calibration procedure [9].

The proposed procedure consists of the following steps:

A. Build a decision tree (D-Tree) In this step, the possible states of the project are recognized.

They may be different phases or specific stages. The possible decisions that could be made when considering a state are recognized. Making a decision would lead to a transition from one state to another. All possible transitions are identified. The result of this step is the creation of a D-Tree. The considered D-Tree is shown in Fig. 1.

B. Build a lattice of state variables (X-Tree) In this step, quantifiable economic magnitudes which may

depend on the result of the project (state variables) need to be identified. This currently proposed method does not include the correlation between these variables, assuming it does not exist.

The tree starts from a known present value of state variables. Based on the history of changes of this magnitude, the values u and d can be determined in the calibration process. A tree of possible changes in state variables is formed and shows possible scenarios of the situation, presented in Fig. 3. Calibration is an appropriate choice for the number of steps and the choice of parameters d, u, so as to best meet the future value of the variable state.

Figure 3. The binomial tree of state variables

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C. Build a tree of the project values (V-Tree) If a project evaluation is based on many state variables, it is

therefore presented in a vector of values. When considering two state variables, the V-Tree grows in two dimensions:

+

+

+

=

)1n,i,i(V),n,i,i(X(f...

)1n,i,i(V),n,i,i(X(f))1n,i,i(V),n,i,i(X(f

)n,i,i(

21s

21ss

212

2122

211

2111

21mV (1)

We denote the present value of the project, which is dependent on two state variables, as:

V(s)(i, j, n) - utility value of project in period n, with i - number of falls of first state variable and j - number of falls of second state variable

The calculation of the value of the V-Tree starts from the end (final value). We assume that the final value of the project is a function of state variables:

=

)N,j(X(f)N,i(X(f

)N,j,i(m22

11V (2)

On this basis, the remaining values of the V-Tree are successively calculated. Trees are constructed for each state of the project. The calculation of value is done by backward induction. Based on the value of the project after its completion (which is usually equal to the state variable, or the right formula for this variable is calculated), the values of the project in the preceding nodes are calculated.

We will consider two possibilities:

in the case of financial factors, when the present value is the discounted expected value of subsequent values

( ) tr11d11u1 e)1n,j,1i(V)1n,j,i(V)n,j,i(V ++++= (3)

( ) tr22d22u2 e)1n,1j,i(V)1n,j,i(V)n,j,i(V ++++= (4)

Figure 4. The V-Tree

in the case of other factors, when the present value is the expected value of subsequent values

)1n,j,1i(V)1n,j,i(V)n,j,i(V 11d1

1u1 ++++= (5)

)1n,1j,i(V)1n,j,i(V)n,j,i(V 22d2

2u2 ++++= (6)

Previous values are weighted by the probability of achieving those values, which are also derived from the data.

D. Determine effective transitions (decisions) The values determined by the formulas (3) (6) must be

calculated for each decision, so we have a superscript denoting the decision on the value achieved:

=

)n,j,i(V

)n,j,i(V)n,j,i( A

AA

2

1V (9)

for decision Act and

=

)n,j,i(V

)n,j,i(V)n,j,i( W

WW

2

1V (10)

for decision Wait.

Assuming that the project is properly managed, a favorable decision will be chosen.

Amm

Wmm

Am

Wm

;Act

;Wait)n,j,i()n,j,i(

VV

VVVV

=

=

ELSE

THEN IF (11)

In (11) the preference between the two vectors must be checked. There are many methods for determining the preferred decision based on the preferences of the decision maker. The simplest solution is used in a dynamic programming scalarization approach [19]. The resulting vector components are weighted coefficients indicating preferences. Single numbers are thus obtained, which can be directly compared.

If we denote:

=k

Wkk

Wm )n,j,i(Vw)n,j,i(Q (12)

=k

Akk

Am )n,j,i(Vw)n,j,i(Q (13)

where wk 0 are weights of assessments,

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the calculations are simplified to:

AttemptWait)n,j,i(Q)n,j,i(Q Am

Wm

ELSE THEN IF >

(14)

After determining that the weights of the criteria are sufficient, the two values are arithmetically compared in order to determine the desired decision.

IV. NUMERICAL EXAMPLEWe will consider a six-month project of a social nature. The

project may begin in the first or second half of 2013. Its cost is 400 thousand PLN. After its implementation, financing in the amount of 100 thousand euros will be obtained. In addition, if the project proves to be purposeful, continued co-operation is possible with the financing institution. The purposefulness of the project depends on the development of the level of unemployment. If the level becomes high, the project will be purposeful. If the unemployment rate drops, its implementation will be useless.

A. Build a decision tree (D-Tree) The project can be launched at the beginning of 2013 or in

July 2013. The decision tree is shown in Fig. 1.

B. Build a lattice of state variables (X-Tree) In this example, we have two state variables, X1 - the

EUR/PLN exchange rate and X2 the unemployment rate. These were evaluated based on historical data. Scenarios of the evolution of these indicators as an X-tree are shown in Tables I and II.

TABLE I. X-TREE FOR EUR/PLN EXCHANGE RATE

n X1i 0 1 2 0 4.0946 4.1548 4.2150

1 4.0344 4.0946

2 3.9742

TABLE II. X-TREE FOR UNEMPLOYMENT RATE

n X2i 0 1 2 0 10.4 10.8 11.2

1 10.0 10.4

2 9.6

C. Build a tree of the project values (V-Tree) According to (2), we define values as in the end nodes:

For the first state variable, the end value is equal to 100 thousand EUR multiplied by X1, minus costs equal to 400 thousand PLN if the project was completed. The end value is equal to 0 if the project was not completed.

For second state variable, the value is equal to 10.0 if the unemployment rate is higher than 10% and 0.0 if is less than 10 %.

The calculated values are shown in the first row and the last column of Table III. Previous values were calculated from (3) for the EUR/PLN exchange rate and from (6) for the unemployment rate. Probabilities for state variables, calculated from historical data, are equal:

For the first state variable, the EUR/PLN exchange rate, we have 1u = 0.447 and 1d = 0.553

For the second state variable, the unemployment rate, we have 2u = 0.625 and 2d = 0.375

The value tree shown in Fig. 4 is presented in Table III. The calculations assumed a risk free rate r equal to 4%.

TABLE III. V-TREE

n 0 1 2

Act -Wait Act -Wait

10

515.

10

515.

10

53.

10

53.

10

521.

10

59.

10

521.

10

59.

10

521.

10

59.

0

521.

059.

10

59.

10

62.

10

59.

10

62.

10

59.

10

62.

059.

062.

10

78.-

6887

.

.

10

514.- 00

256514

..

- 00

10

72.- 00

25672

..

- 00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

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D. Determine effective transitions (decisions) Determining the effective decision after the first period is

obvious. All evaluations of the project for the decision Wait are dominated by the decision Act. It is preferable in this situation to begin the project. Also, at the beginning of the year, an estimated project value vector for the decision Act dominates the calculated vector for the decision Wait. In this situation, the only right decision is to start the project at the beginning of the year.

V. CONCLUSIONSThis paper presents the outline of the valuation method of

development projects in which there are real option situations. The proposed method takes into account the dependence of the project as a result of two independent random factors, which are called state variables. It is based on binomial trees and uses multi-criteria dynamic programming. Its use should allow not only the possibility to make the right decisions about a project, but also the possibility to support decision making during its implementation.

The presented method shows a high sensitivity to the adopted final value function. In the case of economic factors, the form of this function is obvious, whereas in the case of other factors, this is a utility function. Determining this function should be the subject of further research.

ACKNOWLEDGMENTThis research was supported by the grant from National

Science Centre Poland (NCN), project number NN 111 477740.

REFERENCES

[1] D. Bednyagin, E. Gnansounou, Estimating spillover benefits of large R&D projects: Application of real options modelling approach to the case of thermonuclear fusion R&D programme, Energy Policy, vol. 41, 2012, pp. 269-279.

[2] F. Black, M. Scholes. The Pricing of Options and Corporate Liabilities., The Journal of Political Economy, vol. 81, No. 3, 1973, pp. 637-654.

[3] P.P. Boyle, Options: a Monte Carlo approach., Journal of Financial Economics, vol. 4, 1977, pp. 323-338.

[4] P.P. Boyle, A Lattice Framework for Option Pricing with Two State Variables., Journal of Financial and Quantitative Analysis, vol. 23, No. 01, 1988, pp. 1-12.

[5] Ch. Y. Chang, Critiques of the Existing Techniques for the Evaluation of Renewable Energy Projects, 2012 IEEE International Technology Management Conference, 2012, pp. 151-158.

[6] S.D. Comello, M.D. Lepech, B. Schwegler,"Project-Level Assessment of Environmental Impact: Ecosystem Services Approach to Sustainable Management and Development", Journal Of Management In Engineering, vol 28, No. 1, 2012, pp. 5-12.

[7] .J.C. Cox, S.A. Ross, M. Rubinstein, Option pricing: a simplified approach., Journal of Financial Economics, vol. 7, 1979, pp. 229-263.

[8] A.K. Dixit, R.S. Pindyck, Investment under Uncertainty. Princeton University Press, Princeton, 1994.

[9] G. Guthrie, Real Options in Theory and Practice. Oxford University Press, Oxford 2009.

[10] J. Ingersoll, S.A. Ross, Waiting to Invest: Investment and Uncertainty, The Journal of Business, vol. 65, No. 1, 1992, pp. 1-29.

[11] R.L. Kumar, Managing risks in IT projects: an options perspective, Information & management, vol. 40, No. 1, 2002, pp. 6374.

[12] J. Mun, Real Options. [in] J. B. Abrams: Quantitative Business Valuation. Wiley. 2nd ed. Hoboken, 2010.

[13] S.C. Myers, Interactions of Corporate Financing and Investment Decisions - Implications for Capital Budgeting., The Journal of Finance, vol. 29, No. 1, 1974, pp.1-25.

[14] Office of Government Commerce: Managing Successful Projects with PRINCE2 (2009 ed.). TSO, 2009.

[15] E. Plottu, B. Plottu, Total landscape values: a multi-dimensional approach, Journal Of Environmental Planning And Management, vol. 55, No 6, 2012, pp.797-811.

[16] S. C. Rambaud, M. J. M. Torrecillas, Some considerations on the social discount rate, Environmental Science & Policy, vol. 8, No. 4, 2005, pp. 343-355.

[17] R.U. Seydel, Tools for Computational Finance, Fourth ed, Universitext. Springer, Berlin Heidelberg, 2009.

[18] L. Trigeorgis, Real Options and Interactions with Financial Flexibility., Financial Management, vol. 22, No. 3, 1993, pp.202-224.

[19] T. Trzaskalik, Multiobjective Analysis in Dynamic Environment, Publisher of The Karol Adamiecki University of Economics in Katowice, Katowice 1988.

[20] C.P. Heidkamp, "A theoretical framework for a 'spatially conscious' economic analysis of environmental issues", Geoforum, vol. 39 No.1, 2008, pp. 62-75.

[21] R. Evans, C. J. Moran, D. Brereton, "Beyond NPV - A review of valuation methodologies and their applicability to water in mining", Proceedings of the Water in Mining 06 Conference, Australasian Institute Of Mining And Metallurgy Publication Series, 2006, pp. 97-103.

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