L. Dimova, P. Sevastjanov,* D. Sevastjanov** icon

L. Dimova, P. Sevastjanov,* D. Sevastjanov**

НазваниеL. Dimova, P. Sevastjanov,* D. Sevastjanov**
Дата конвертации26.09.2012
Размер91.25 Kb.


L. Dimova, P. Sevastjanov,* D. Sevastjanov**

* Institute of Comp.& Information Sci., Technical University of Czestochowa,Dabrowskiego 73, 42-200 Czestochowa, Poland sevast@icis.pcz.czest.pl

** Maltex.com, inc., 100 William st., 6 floor, New York, NY, USA,



Capital budgeting is based on the analysis of some financial parameters of considered investment projects. It is clear that estimation of the investment efficiency, as well as any forecasting, is rather uncertain problem. In the case of stock investment we can predict to some extent the future benefits using stock's history and statistical method but for a small time horizon only. In real active investment we usually deal with the business-plan which takes a long time-as a rule some years- for its realisation. In such cases, description of uncertainty within the framework of traditional probability methods usually is impossible because of absence of an objective information about future events probabilities. That is why, during the last two decades the growing interest to the application of interval and fuzzy methods in budgeting has being observed. In the present article the technique for the fuzzy- interval evaluation of financial parameters is presented. This allows to use with the greater completeness than traditional methods a priori uncertain information about future cash flows, interest rates. As the result the technique allows us to obtain the fuzzy interval and weighted non-fuzzy values for the main financial parameters NPV and IRR, as well as the quantitative estimation of investment’s risk.

The other problem is that usually we must consider a set of different local criteria based on the financial parameters of investments. As one of its possible decision, the numerical method for optimisation of future cash-flows in fuzzy setting when considering the generalised project's quality criterion as the compromise between local criteria of a profit maximisation and financial risk minimisation is proposed.

^ Keywords: Capital Budgeting;Risk; Fuzzy Sets; Multiobjective Optimisation

  1. Introduction

At first, let us consider the usual non-fuzzy approaches to the capital budgeting problem. There are a lot of financial parameters proposed in literature[1-4] for budgeting .

The main of them are: the Net Present Value (NPV), Internal Rate of Return (IRR), Payback period(PB) , Profitability Index (PI). These parameters are usually used for project quality estimation but in practice they have deferent importance.

It is earnestly shown in [5] that the most important parameters are NPV and IRR (see Table. 1).

Therefore, our further consideration will be based on the analysing only NPV and IRR. Good review of other useful financial parameters is in [6].

Table 1. Frequency of the financial parameters using (from 103 largest petroleum and gas companies of USA in 1983)

Type of using









Other parameters



Net Present Value is usually calculated as follows:

, (1)

where d is a discount rate; tn is the production starting year; tc is the year of the ending of investments; KVt is a capital investment in a year t, Pt- is an income in a year t, T is a duration of investment project in years.

Usually discount rate is equal to the average bank interest rate in investor's country or equal to other appropriate value for estimation of return in the case of alternative capital investments to the other projects or securities.

The economic nature of internal rate of return (IRR) can be explained as follows. As an alternative to investments in the analysing project the depositing under some bank interest (distributed in time as in the case of the analysing project) is considered. It is suggested that all the incomes that will be received during realisation of project will also be deposited with the same interest rate. If discount rate is equal IRR, then the investment in the project will give the same total income, as in the case of depositing. Thus, both alternatives are economically equivalent. If actual bank discount rate is less then IRR of the considered project, the investment in the project is more preferable. Therefore IRR is the threshold discount rate selecting effective and the ineffective investment projects. The value of IRR is the solution of the non-linear equation with respect to d:


Estimation of IRR is frequently used as a first step of financial analysis. Only projects with IRR which are not below than some accepted threshold value (usually 15-20 %) are choosing for further consideration.

Nowadays, traditional approach to evaluation of NPV, IRR and other financial parameters is subjected to the quite deserved criticism, since the future incomes Pt, capital investments KVt and rates d are rather uncertain parameters. Uncertainties which we meet in capital budgeting, differ from ones in the case of share prices forecasting and can not be adequately described in probability terms. In real active investment we usually deal with the business-plan which takes a long time-as a rule some years- for its realisation. In such cases, description of uncertainty within the framework of traditional probability methods usually is impossible because of absence of an objective information about future events probabilities. Thus, what we really have in such cases are some expert's estimations. In real world situations, investors or experts involved are able to predict confidently only intervals of possible values Pt, KVt and d and the most expected values inside these intervals. That is why, during last two decades the growing interest to the applications of interval arithmetic [7] and fuzzy sets theory methods[8] in budgeting has being observed.

After pioneer works of T.L.Ward [9] and J.U. Buckley [10], some other authors contributed to the development of fuzzy capital budgeting theory [11-24]. We can say now that almost all problems of fuzzy NPV estimation are solved but an interesting and important problem of project's risk assessment on the base of fuzzy NPV becomes a priority one.

Unresolved problem is the fuzzy estimation of IRR. Ward [9 ]considers Eq.(2) and explains that such a expression cannot be applied for the fuzzy case because the right hand side of f Eq.(21 ) is fuzzy, 0 is crisp, and an equality is impossible. Hence, the Eq. (2) is senseless in the fuzzy setting.

In [22] the method for fuzzy IRR estimation is proposed where -cut representation of fuzzy numbers [25] has been used. The method it based on the assumption [see [22] p. 380] that set of equations for IRR determination on each -level may be presented as (in our notation)

, (3)

where , i=0 to n , are crisp interval representations of fuzzy cash flows on - levels. Of course, from linear equations (3) all crisp intervals expressing the fuzzy valued IRR may be obtained. Regrettable, there is a little mistake in (3). The right crisp interval representation of Eq. (2) on - levels must be written as

. (4)

There is no way to get an interval decisions for IRR from (4) but real ones may be obtained (see Section 3, below).

Another problem not presented in literature is optimisation of cash flows in fuzzy setting.

The rest of paper is set out as follows. In Section 2 we recall the method for fuzzy estimation of NPV and introduce the risk estimation based on conception of degree of fuzziness. Section 3 describe the method for real value solving Eq.(2) in the case of fuzzy cash flows. The set of useful real valued parameters connected with real value solving of Eq.(2) are proposed and analysed, too. In Section 4, the numerical method for optimisation of cash flows as the compromise between local criteria of a profit maximisation and financial risk minimisation is proposed.

  1. ^ Fuzzy NPV and risk assessment connected with it

The technique offered is based on fuzzy extencion principle [8]. Thus, the values of uncertain parameters Pt, KVt and d are substituted by corresponding fuzzy intervals. In practice it means that the expert sets lower - Pt1 (pessimistic value) and upper - Pt4 (optimistic value) boundaries of intervals and internal intervals of the most expected values [Pt2, Pt3] for the parameters analysing (see Fig. 1).. The function (Pt) is usually interpreted as a membership function, i.e. the degree to which the parameter’s values belong to considered interval (in our case [Pt1, Pt4]). The membership function changes continuously from 0 (area out of interval) up to maximum value equal to 1, in area of the most possible values. It is obvious that the membership function is the generalisation of usual set’s characteristic function which is equal to 1 for all values of parameters inside a set, and is equal to 0 in all other cases.



^ Figure 1. Fuzzy interval of uncertain parameter Pt and its membership function (Pt ).

The linear character of function is not obligatory, but such a mode is the most used and allows to represent fuzzy intervals is convenient form by quadruple Pt = {Pt1, Pt2, Pt3, Pt4}. Then all the necessary calculations are carried out using the special fuzzy- interval arithmetic rules.

Let us recall some basic principles of fuzzy arithmetic [25].

In general, for arbitrary forms of membership functions the technique of fuzzy- interval calculations is based on the representation of initial fuzzy intervals by so-called -cuts (Fig.1) that, in fact, are crisp intervals associated with the corresponding degrees of a membership. All further calculations are made with those -cuts according with the well known crisp interval-arithmetic rules and the resulting fuzzy- intervals are obtained as a disjunction of corresponding final -cuts being calculated.

Thus, if f ^ A is a fuzzy number, then

where A is the crisp interval {x: A (x) },  A is fuzzy interval {( x,): x A }. Thus, if A, B, Z are fuzzy numbers (intervals) and @ is an operation from {+, -, *, /} then

Z = À@Â=. (5)

Since in the case of -cut presentation, fuzzy arithmetic is based on crisp interval arithmetic rules, the basic definitions of applied interval analysis must be presented too. There are some definitions of interval arithmetic ( see [26,27]), but in practical applications the so-called «naive» form proved the best. According to it, if А = [a1, a2] and В = [b1, b2] are crisp intervals, then

Z = А@В={ z=x@y, }. (6)

As the direct consequence of the basic definition (6) the next expressions were obtained:

А+В=[a1+b1, b2+b2], А-В=[a1-b2, a2-b1],

А·В=[min(a1·b1, a2· b2, a1·b2, a2·b1), max(a1·b1, a2· b2, a1·b2, a2·b1)],

А/В=[a1, a2] · [1/b2, 1/b1]

Of course, there are many internal problems within applied interval analysis, like the division by zero-containing interval, but in general it can be considered as the good mathematical tool for modelling under the conditions of uncertainty.

To illustrate, let us consider an example. Let we have the investment project, in which building phase proceeds two years with the investments KV0 and KV1 accordingly. The profits are expected only after finishing building phase and will be obtained during two years (P2 and P3). It is suggested that fuzzy interval for discount d remains stable during the time of project realisation. The appropriate trapezoidal initial fuzzy intervals were as follows: KV0 = {2,2.8,3.5,4}; KV1 = {0,0.88,1.50,2}; KV2 = {0,0,0,0}; KV3 = {0,0,0,0}; P0 = {0,0,0,0}; P1 = {0,0,0,0};P2 = {6.5,7.5,8.0,8.5}; P3 = {5.5,6.5,7.0,7.5}; d = {0.08,0.13,0.22,0.35}.

The resulting fuzzy interval NPV, calculated using fuzzy extension of Eq.(1) is presented in Fig. 2.

Figure 2. Resulting fuzzy interval NPV.

The obtained fuzzy interval allows to estimate the boundaries of possible values of predicted NPV, the interval of most expected values, and also- that is very important- to evaluate a degree of financial risk of the investments.

To estimate the financial risk, we have taken into account the following inherent property of fuzzy sets. Let A be some fuzzy subset of ^ X, being described by the membership function (A). Then complementary fuzzy subset A has the membership function (A)=1-(A). The principal difference fuzzy subset from usual precise ones is that intersection of fuzzy A and A is not empty, that is AA= B, where B is a not empty fuzzy subset, too. It is clear that the closer A to A, the more power of a set B and more A differ from ordinary sets.

Using this circumstance R Yager [28] proposed a set of grades of nonfuzziness of fuzzy subsets


Hence, the grade of fuzziness may be defined as


The definition (8) is in compliance with the obvious requests to the grade of fuzziness. If A is a fuzzy subset on X , (A) is its membership function and dd is the corresponding grade of fuzziness, then following properties should be observed:

1) dd(A) = 0, if ^ A is crisp subset;

2) dd(A) has a maximum value if (A) = 1/2 for xX;

  1. dd(А1)>dd(A), if (х)<(y) (xA1; yA).

It is proved that the introduced measure is similar to the Shannon entropy measure [28].

In the most useful case (p = 1) expression (8) is transformed to

. (9)

It is clear (see Eq.(9)) that grade of fuzziness is rising from 0 if (A) = 1 (crisp subset) up to 1 if (A) = 1/2 (maximum degree of fuzziness).

With respect to our problem the grade of nonfuzziness of fuzzy interval NPV can linguistically be interpreted as a risk or uncertainty of obtaining the Net Present Value in the interval [NPV1, NPV4]. Really, the more precise, (more «rectangular») interval we receive, the more degree of uncertainty and risk we obtain. Of course, at first this assertion seems paradoxical. However, any precise (crisp) interval is not containing any additional information about relative preference of values placed inside it. Therefore, it contains less useful information, than any fuzzy interval being constructed on its basis. In the latter case the additional information that reducing uncertainty is derived from the membership function of considered fuzzy interval.

Thus, the approach proposed for evaluation of NPV inevitable generates two criteria for estimation of the future profits: fuzzy interval NPV and degree of its uncertainty (degree of risk).

Therefore, problem investments efficiency evaluation on the base of NPV becomes two-criteria and requires the special approach and appropriate technique. Recently, we proposed such technique [29] based on the fuzzy set theory was developed , however its detailed consideration is out of scope of this paper.

  1. ^ The set of real value IRR estimations based on the fuzzy cash flows

In essence, problem of Internal Rate of Return (IRR) evaluation looks as fuzzy interval solution of the Eq.(2) with respect to d.

It is proved that the solution of the equations with fuzzy parameters is possible by expression of these parameters (Pt , KVt and d in our case) as a sets of corresponding  - cuts. As the result for the problem of evaluating IRR we obtain a system of the non-linear crisp - interval equations:

, (10)

where [Pt] , [KVt] and [d]a are the crisp intervals on corresponding - cuts.

In general, we can state that the naiv assumption that in a right hand side of Eq.(10) there should be a degenerated zero interval [0,0], does not ensure deriving of an adequate outcomes since there is non-degenerated interval expression on the left hand side of Eq. (10), but let us consider this situation thoroughly.

As the simplest example, consider two-years project, when all investments are finished in the first year, and the production and deriving of the incomes begins and ends in the second year. Then each of the equations for -cuts (10) should be divided on two :

- left boundary of an interval NPV


- right boundary of an interval NPV

The formal solution (10) with respect to d1 and d2 is trivial: , however it is senseless, as the right boundary of an interval [d1, d2] always appears less than left one.

This, on the first glance, absurd result should be easily explained from common methodological positions. Really, the rules of interval mathematics are constructed in such away that any arithmetical operations with intervals give us as an interval, too. These rules are in the full correspondence with the well known common methodical position, according to which any arithmetical operation with uncertainties must to increase the total uncertainty and the system’s entropy . Therefore, if in our case we place to the right hand side of (10) and (11) a degenerated zero intervals, it will be equivalent to the request of reducing uncertainty of the left parts up to zero that should be possible only in the case of inverse character of an interval [d1, d2] that in turn can be interpreted as a request of entering negative entropy in a system.

Thus, the presence a degenerated zero interval in the right hand sides of the interval equations is incorrect. The more acceptable approach to solving of this problem has been constructed with the help of following reasons. It is easy to see, when analysing expressions (11) that for any value d1 the minimal width of an interval NPV is reached if d2 = d1. This is in accordance with common methodical positions: the minimum uncertainty of an outcome (NPV) is reached in the case of minimum uncertainty of all the system’s parameters used. It is clear (see Fig. 3 ) that the most reasonable decision of «zero» problem is the request for the middle of an interval NPV to be placed on a zero point (request of symmetry of an interval concerning to zero).

The obvious, on the first sight, intention to minimise the sizes of received interval NPV results in deriving positive or negative intervals of minimum width, but not intersecting an zero point, that does not correspond to the natural definition of a zero containing interval. Besides it can be easily proved that only request of symmetry of a zero containing interval ensures an asymptotically valid outcome in the case of contraction of the boundaries of all considered intervals to their centres.

Thus, generally problem is reduced to searching such exact (non-interval) values d, which can provide a symmetry with respect to zero resulting intervals NPV on each -cut in the equations (10), i.e. would guarantee fulfilment of a request (NPV1 + NPV2) = 0, for each = 0,0.1,0.2..., 1.

Of course, the problem is decided by the numerical methods.

Figure3. The discount dependence of an interval NPV when the investments in the first year is KV0 = [1,2], income in the second year is P1 = [2,3]: and D (NPV) is width of an interval NPV.

To illustrate the above theoretical considerations, let's compare two investment projects that must be realised during 4 years . Fuzzy cash flows Kt = Pt- KVt are defined with the help of four-reference points form described above(see Table 2). It worth noting that the data of first project are more certain.

Table 2.

Fuzzy parameters of compered projects.






































The results of estimations for two comparing investment projects with different fuzzy cash flows are presented on Fig. 4., too. It is seen that values of IRR obtained for each -cut can increase or decrease with growing of and as the result for each project own set of possible real number values of IRR has been obtained. Thus, the problem of interpretation of the results rises.

To do this, we propose to reduce the sets of IRR obtained on each - cut to the little set of parameters which can be easily interpreted. The first elementary parameter- average value IRRm - is certainly convenient, however it does not take into account that with growing of the reliability of an outcome increases too, i.e. IRR, obtained on higher -cuts are more expected, than obtained on lower ones, because of -cut’s definition. On the other hand, a precise intervals [NPV1, NPV2] corresponding to each of IRR has a widths which being in some sense a measure of uncertainty for received non-interval value IRR, since the widths of intervals [NPV1, NPV2] characterise quantitatively the difference of the left hand side of Eq.(10) from degenerated zero interval [0,0]. This allows us to introduce two weighted estimations of IRR on a set IRR: least expected (least reliable) IRRmin and most expected (most reliable) IRRmax :

, (12)

, (13)

where n is the number of -cuts.

In decision making practice, when choosing the best project, it is worthy to use all three proposed parameters IRRm, IRRmin, IRRmax.

Interpretation of [NPV1, NPV2] as performance for uncertainty of IRR allows to propose the quantitative and expressed in monetary units evaluations of financial risk of project considered (uncertainty degree of received values IRRср, IRRmin, IRRmax as a consequence of the initial data uncertainty ):


Parameter Rm can play a key role in project's efficiency estimation.

For our example we get

Project 1: IRRmin = 0,34; IRRmax = 0,327; IRRср = 0,335; Rср = 1,56.

Project 2: IRRmin = 0,322; IRRmax = 0,329; IRRср = 0,325; Rср = 3,52.

Thus, the considered projects have rather close values of IRRср, IRRmin, IRRmax. At the same time ,the risk value Rcp for the second project is considerably higher than risk of the first one. So, the first project is the bast one.

Except the parameters described above, there are some other useful estimations: IRRnr - most reliable value of IRR - connected with the minimum interval [NPV1, NPV2] nr among of all [NPV1, NPV2] and IRRnr - the least reliable value of IRR - connected with the maximum interval [NPV1, NPV2] r among all of [NPV1, NPV2] . It is clear, that [NPV1, NPV2] nr and [NPV1, NPV2] r are the risk estimations for appropriate IRRnr and IRRr.

It should be noted (see Fig. 4) that difference between values of IRRnr for the projects compered is rather negligible, but the difference in risk estimations is considerable.

^ 4. Method for numerical solution of project optimisation problem

Proposed here approach to optimisation problem is based on the consideration of all the initial fuzzy intervals Pt and KVt as the restrictions on controlled input data and on the assumption that dt is the a random parameter describing an external, in relation to the considered project, uncertainty. We also take into account that there may be some preferences on the interval of possible values of d which may be expressed by certain membership function , say d(d). Thus, we deal with the random as well as with a fuzzy uncertainties when describing discount factor.

The problem is decided in two steps.

At first, according to the fuzzy extension principle we substitute in equation (1) all the parameters Pt , KVt and dt by corresponding fuzzy-intervals. As the result we obtain fuzzy-interval estimation of NPV.

On the next step, we consider the obtained fuzzy-interval NPV as the restriction on the profit that can be derived we built the local criterion of NPV maximisation.

For mathematical description of local criteria we use so-called desirability functions which are, in essence, the special interpretation of usual membership functions. Briefly, the desirability function is rising from 0 (in the field of inadmissible values of its argument) up to 1 (in area of the most preferable argument's values). Thus, the construction of desirability function for NPV is rather obvious: the desirability function NPV (NPV) can be considered only on an interval of possible values restricted by the interval [NPV1,NPV4] and, naturally, the more values of NPV, the more degree of desirability (see Fig. 4).

Figure. 4. Connection between restriction and local criterion:1-intitial fuzzy interval of ^ NPV (fuzzy restriction); 2- the desirability function NPV (NPV) .

The initial fuzzy intervals Pt and KVt, are also considered as the desirability functions ..., which describing the restrictions on the controlled input variables. It is clear that initial intervals were already constructed in such a way that in the case of their interpretation as the desirability functions, the more preferable values from intervals of Pt and KVt appear those which are more realisable (possible). Since these desirability functions are connected with the possibility of realisation of corresponding values of variables Pt and KVt, they describing implicitly the financial risk of the project.

On the set of all the desirability functions the general criterion maximising is created:

, (15)

where 1 and 2 are ranks, characterising the relative importance of profit maximisation and risk minimisation local criteria; is the minimising operation, - desirability function of NPV.

There are many different form of general criterion were using in applications. As emphasised in [30], now choosing of concrete realisation of aggregating operator, which is usually called t-norm, is rather the application dependent problem. However, the choosing min-operator in Eq.(15) is the most straight-out approach when we not permit the compensating small values of some criteria by the great values of other ones. The problem is reduced to searching of non-interval (precise) values of PP1, PP2..., KKV1, KKV2... (changing in the appropriate fuzzy intervals P1, P2..., KV1, KV2), which have to maximise the general criterion (15).

The problem is complicated by fact that the discount d is a random parameter, distributed in a specific interval.

Procedure of a solution was carried out as follows.

At first, from interval of discount’s possible values by random way a fixed number is selected. Further with the help of Nollaw-Furst random method the optimum solution is obtained, as the best compromise between uncertainty of basic data and intention to derive the maximum profit, i.e. the optimisation problem comes to the maximisation of the general criterion (15). The optimal values PPdt and KKVdt, , are a local optimum decision for given discount value. Therefore, we are repeating the described procedure with the new random discount values until the statistically representative sample of optimum solutions at the various d will had been obtained. The final optimum PP0t , KKV0t values are calculated as the weighted evaluations taking into account the degrees of possibility of various di, which are defined by an initial fuzzy interval d. with a membership function d(di)

, (16)

where m is the number of discount values used for the solution of a problem. Similarly , all KKV0t can be expressed, too.

It is possible also to take into account the values of a general criterion in the optimum points:

, (17)

where 1,2 are corresponding weights. The similar expression we have for KKV0t .

It worth noting that last expression enables us to take into account, apart from of reliability of values di, the degree of compatibility (in other words, the degree of consensus) in compromise at each selected values of discount.

Obtained optimal PPt0 and KKt0 may be used for the final project quality estimation.

For the example considered in previous Section (Table 2, project 1), using expressions (16), (17) , we have obtained the results presented in Table 3.

Table.3. The results of optimisation.

Expression (16)

Expression (17)


























Further, by substituting these (PPt0 and KKt0 ) and fuzzy interval d in the expression (1), we get the optimal fuzzy value of NPV.

For our example we obtain

NPV16 = {4.057293, 6.110165, 8.073906, 9.454419},

when using the expression (16) and

NPV17 = {4.065489, 6.109793, 8.064094, 9.436519}

on the base of expression (17).

It is clear that there are no great deference between the results obtained using expressions (16) and (17) in our case.

In Fig. 5, the fuzzy NPV16 obtained with using the optimal PPt0 and KKt0 is compared with the initial one obtained using direct account on the initial fuzzy values Pt and KVt, without use of optimisation. Of course, in optimal case we have a greater mean value of fuzzy interval NPV.

Figure 5. The comparison of the initial and optimal fuzzy intervals NPV

Using optimal PPt0 , KKt0 and method described in Section 2 , the degree of project risk may be estimated , too. This risk can be assumed as the financial risk of the project as a whole..

For the aims of usual accounting practice it is possible to calculate the average weighted value of NPV by use the expression:

. (18)

For our example the NPV16= 6.8931 and NPV17 = 6.8942 have been obtained.

  1. Conclusion

The natural way for project risk assessment is to treat it as the degree of fuzziness of fuzzy valued Net Present Value, NPV. It is clearly shown, why it is impossible to get a fuzzy Internal Rate of Return, IRR. The only real valued IRR may be obtained as a decision of fuzzy equation, but a set of new useful parameters connected with IRR and characterising uncertainty of the problem may be obtained as the additional result.

Multiobjective project optimisation problem in the mixed fuzzy and random setting is formulated as the compromise between the local criteria of profit maximisation and risk minimisation. Numerical method for deciding of this problem is described and tested.


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L. Dimova, P. Sevastjanov,* D. Sevastjanov** iconДокументы
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2. /dimova.doc
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