Build-operate-transfer of airport in fuzzy cost of capital and fuzzy capital budgeting.

ABSTRACT

Most previous studies concerning financial performance evaluation focus merely on the cost control, which might directly influence the survival of a company. This paper attempts to construct a new optimal capital planning decision method. The topic is based on fuzzy capital cost and fuzzy capital budgeting under fuzzy economic scenario. And it is developed focus on BOT (Build-Operate-Transfer) of airport. To efficiently handle the fuzziness of decision variable with respect to planning and decision of optimal capital on airport, the linguistic values, subjectively represented by triangular fuzzy numbers, are used to act as the evaluation tool. To introduce the computational methods of fuzzy weighted average cost of capital (FWACC) and fuzzy modified internal rate of return (FMIRR) are the base of fuzzy capital budgeting proposed in this study. By utilizing the new finance decision method, not only the decision-maker can handle more true information and make the best planning. But also the government decision-maker can make a well decision for airport in BOT under fuzzy economic scenario.

Keywords: Earning management, Risk management, Fuzzy capital budgeting, Fuzzy capital cost, Triangular fuzzy number

1. INTRODUCTION

Since the Asian financial crisis in 1997, the government's expenditures are curtailed. Therefore, the government would like to cooperate with some corporations to reduce its expense. The purpose of this paper is to present some attempts under financial plan on build-operate-transfer (BOT) of airport in fuzzy cost and fuzzy capital budgeting. However, successful packaging of a BOT proposal means getting all of the political, technical, commercial and financial elements of a project put together in a proposal so that adequate funds can be committed and advanced to the project company and construction can begin (Tiong and Alum, 1997). Therefore, we construct a new optimal financial planning decision method based on fuzzy capital cost and fuzzy capital budgeting focused on the BOT (Build-Operate-Transfer) of an airport.

BOT is a deal between the government and corporations. The government hands over the right of build and operation of an airport to corporations. After 30 or 50 years, the airport has to be transferred to the government for free. The entrepreneurs must take a long view in such financial contract, because of the long duration and high capital costs of infrastructure projects and changing priority of the host governments. Upon successful construction of the projects, the actual concession period can last for 3050 years depending on the type of project. During these operational phases, the maintenance of throughput in line with original forecast remains an area of great uncertainty and challenge to the entrepreneurs (Yeo and Tiong, 2000). How the corporation establishes a good financial evaluation planning under uncertain environment on the cost and budget control before investment on an airport is an important issue and deserves a through investigation in its own right. Thus, this paper studies the cost capital and capital budgeting of an airport that corporation can establish a good financial plan in order to reduce investment risk.

Variables of cost control and budgeting management are uncertain. And for maintenance of throughput in line with original forecast remains, we should not only view a forecast point. There are many fields of science treating uncertainty through the concepts and methods of probability theory. There are many situations in which the source of uncertainly is not a random variable but can be represented the form of linguistic variables. The main reason is that result is a very precision value. It could not represent to fluctuate according to the economic situation. In 1965 L. A. Zadeh introduced the fuzzy theory is to grasp the uncertainty in information. Fuzzy numbers are used to quantify the inexact information such as "around," "very," "little," and so on. These fuzzy numbers allow us to manipulate inexact knowledge using mathematical operators and programming techniques (Zadeh, 1975). Thus, this paper will use fuzzy method to resolve in the build-operate-transfer of an airport.

2. LITERATURE REVIEW

2.1 Related Research Issues in Fuzzy Finance

Turtle, Bector and Gill (1994) studied on the using of fuzzy logic in corporate finance. An example of a multinational cash flow netting problem was studied. Uncertain cash flows from subsidiaries make this problem difficult to be specified in a traditional crisp environment. They provided feasibile ranges and the shadow price for changes within the feasible range. Beside, they were successful linking sensitivity analysis and fuzzy logic.

Korvin, Strawer and Siegel (1995) engaged a study in the area of cost variance analysis. Cost accountants must continually be in a good sense and provide professional judgment in the accounting process to deal with the ambiguity of the cost. The use of fuzzy sets to build fuzzy control systems provides a method to incorporate the ambiguity into expert systems.

Chiu and Park (1998) studied on the capital budgeting decisions with fuzzy projects. The authors proposed a capital budgeting model under uncertainty in which the cash flow information was specified as a special type of fuzzy numbers-triangular fuzzy numbers.

Sanchez, Liao, Ray and Triantaphyllou (1999) gave an example for dealing with the impreciseness of future cash flows during the selection of economic alternatives. That paper presented an approach of how to deal with the imprecision of future cash flows. They represented the cash flows as triangular fuzzy numbers. The following figure is showed the relationship among previous related researches.

2.2 Related Research Issues in BOT

Tiong, Robert and Alum (1997) primarily concerned with the evaluation of tender proposals for BOT projects particularly in Asia-Pacific regions. It describes the selection process and examines the current evaluation practices and techniques that based on the net present value (NPV) method, the scoring system and the Kepnoe-Tregon decision-making technique. (The major elements in Kepnoe-Tregon decision-making technique consist of the evaluation statement, the MUST criteria; they responses were solicited from BOT practitioners to establish the major criteria that are commonly used by governments in evaluating BOT proposals. It concludes that government's evaluation goal should be to select a balanced proposal that is financially attractive and technically cost-effective.

Tiong, Robed and Alum (1997) presents an analysis of the issues related to financial commitments of funds required from the lenders and promoters in a BOT tender. It wants to investigate the extent of importance of a high level of financial commitments in a BOT tender, and whether the level of financial commitments provides the competitive advantage and increases the chances of success in a competitive BOT tender. Finally, it recommends the strategies required for developing a successful and competitive financial proposal for a BOT project. It concludes concentrate on three key areas in the process. (A) Developing the financial framework and the financing plan. (B) Developing the financial framework structure and financial risks. (C) And Develop formulation of financial strategies.

Yeo and Tiong (2000) proposed a risk reduction strategy in winning and managing BOT concession. Examples of successful and unsuccessful BOT projects are selectively used to illustrate the elements of the framework. The proposed soft systems methodology process encourages proactive management of BOT negotiation and concession, especially in the positive management of differences and control of variations, and the problem solving capability of an entrepreneurial promoter.

2.3 Related Research Issues in WACC

Babusiaux and Pierru (2001) point out a firm using a discount rate defined at the corporate scale as a WACC may have to value projects subject to a different tax rate from the one used to calculate its discount rate. To determine the economic value of a project, the WACC and Arditti-Levy methods need to be adjusted if the firm allocated to this project a loan representing proportionally more (or less) than the fraction corresponding to the target debt ratio defined by the firm for the projects, in the same class or risk. First, it proposes a method that corresponds to the adjustment of standard WACC calculations. The formulation adopted ("generalized ATWACC method") has the advantage of being independent of any consideration related to debt ratios.

Cigola and Peccati (2003) point out the valuation of levered investment in the practice is made with the WACC approach, even if the superior technique of APV is available. It shows that the APV can be interpreted as the arbitrage free value of the portfolio made by an investment and a supporting loan. Therefore the WACC evaluation allows for arbitrage.

The economic environment is uncertain they would like to get good answer, me too. We use fuzzy method to solve what the project could not display clearly on cost, income and risk.

3. METHODS

We will define fuzzy set theory, triangular fuzzy numbers, calculation fuzzy [alpha]-cut, fuzzy membership range, Fuzzy Weighted Average Cost of Capital (FWACC) and Fuzzy Modified Internal Rate of Return (FMIRR). And incorporate fuzzy numbers into the financial analysis of BOT of an airport.

3.1 Fuzzy Set Theory

Let X be the set of universe, a fuzzy set A in X is characterized by a membership function [f.sub.A](x) which is associated with each element in X a real number in the interval [0, 1], with the value of [f.sub.A](x) representing the "degree of membership" of [x.bar] in A. Thus, the closer the value of [f.sub.A](x) to the unity the higher is the degree of membership of [x.bar] in A. When A is a crisp set, its membership function can take on only two values 0 and 1 representing that an element does not or does belong to A correspondingly (Zadeh, 1965).

3.2 Triangular Fuzzy Number

A fuzzy number A (Dubois and Prade, 1978) in R (real line) is a triangular fuzzy number whose membership function [f.sub.A] [??] R [right arrow] [0, 1] is defined as:

f [sub.A.(x) = {(x - c)/(a - c), c [less than or equal to] x [less than or equal to] a, (x - b)/(a - b), a [less than or equal to] x [less than or equal to] b, 0, otherwises

Where -[[infinity].bar]<[c.bar][a.bar][less than or equal to][b.bar]<[[infinity].bar] A triangular fuzzy number [A.bar] can be represented by (c.bar], [a.bar], [b.bar], the triplet as show in Figure 1. Triangular fuzzy number [A.bar] has a maximum degree of membership on [a.bar]. i.e. [[f.bar].sub.[A.bar]](a)=1. In addition, [c.bar] and [b.bar] are the lower and upper bounds of the support of A which is used to reflect the fuzziness of the spread of the uncertainty. The narrow the interval [[c.bar], [b.bar]] is, the less uncertain the A is.

[FIGURE 1 OMITTED]

3.3 The Algebraic Operations of Fuzzy Numbers based on the a-Cut Concept

The [alpha]-cut of the fuzzy number A is defined as [A.sup.[alpha]] = { x [member of] X [greater than or equal to] [f.sub.A] (x) [greater than or equal to] [alpha], 0[less than or equal to][alpha][less than or equal to]1} = [[A.sub.l.sup.[alpha]], [A.sub.u.sup.[alpha]]]. The A and B are positive fuzzy numbers, i.e., [A.sub.l.sup.[alpha]]>0, [B.sub.l.sup.[alpha]]>0, for all [alpha][member of][0, 1]. Let [A.sup.[alpha]] = [[A.sub.l.sup.[alpha]], [A.sub.u.sup.[alpha]]] and [B.sup.[alpha]] = [[B.sub.l.sup.[alpha]], [B.sub.u.sup.[alpha]]. According to extension principle (Zadeh, 1965) and vertex method (Dong and Shah, 1987), the algebraic operations of any two positive fuzzy numbers [A.bar] and [B.bar] can be expressed as:

Fuzzy addition: [(A[??]B).sup.[alpha]] =[[A.sub.l.sup.[alpha]] +[B.sub.l.sup.[alpha]], [A.sub.u.sup.[alpha]] + [B.sub.u.sup.[alpha]]]

Fuzzy subtraction: [(A[??]B).sup.[alpha]]=[[A.sub.l.sup.[alpha]]-[B.sub.u.sup.[alpha]], [A.sub.u.sup.[alpha]]-[B.sub.l.sup.[alpha]]]

Fuzzy multiplication: [(A [??] B).sup.[alpha]] =[[A.sub.l.sup.[alpha]] [B.sub.l.sup.[alpha]], [A.sub.u.sup.[alpha]] [B.sub.u.sup.[alpha]]])

Fuzzy division: [(A[??]B).sup.[alpha]] =[[A.sub.l.sup.[alpha]]/[B.sub.u.sup.[alpha]], [A.sub.u.sup.[alpha]]/[B.sub.l.sup.[alpha]]]

3.4 Ranking of Fuzzy Numbers

A fuzzy set can be expressed in terms of the concept of a-cut without resorting to the membership function (Terano, Asai and Sugeno, 1991). Thus, we use the [alpha]-cut method to sort fuzzy numbers. Let [A.sub.1],[A.sub.2], ..., [A.sub.i] ..., [A.sub.n], be n fuzzy numbers, and the left and right membership function of fuzzy number [A.sub.i] are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Suppose that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the inverse functions of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , respectively. Define the left integral value [I.sup.L]([A.sub.i]) and right integral value [I.sup.R]([A.sub.i]) of [A.sub.i] as (Liou and Wang, 1992; Yager, 1981):

Let [alpha] [member of] [0,1], j=0, 1, 2, ..., k, and 0 = [[alpha].sub.0] < [[alpha].sub.1] < ... <[[alpha].sub.j]< ... <[[alpha].sub.k] = l, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [??][alpha].sub.j]=[[alpha.sub.j]-[[alpha].sub.j-1, in this paper the ranking value R(A) of fuzzy number A is defined as

R(A)=[[I.sup.R](A) + [I.sup.L](A)]/2. Define the ranking of the fuzzy number [A.sub.i] and [A.sub.j] based on the following rules:

[A.sub.i] > [A.sub.j] [??] R([A.sub.i]) > R ([A.sub.j])

[A.sub.i] < [A.sub.j] [??] R([A.sub.i]) < R ([A.sub.j])

[A.sub.i] = [A.sub.j] [??] R([A.sub.i]) = R ([A.sub.j])

3.5 Fuzzy Weighted Average Cost of Capital (FWACC)

The cost of capital used to analyze capital budgeting decisions should be the company's required return on equity. However, most firms raise a substantial portion of their capital as long-term debt, and many also use preferred stock. For these firms, the cost of capital must reflect the average cost of the various sources of long-term funds used, not just the firms' costs of equity.

Assume that corporation a 10% cost of debt, a 10.3% cost of special equity and a 13.4% cost of equity. Further, assume that corporation has made the decision to finance next year's projects by selling debt. The argument is sometimes made that the cost of capital for these projects is 10 percent, because only debt will be used to finance them.

However, this position is incorrect. If corporation finances a particular set of projects with debt, the firm will be using up some of its potential for obtaining new debt in the future. As expansion occurs in subsequent years, corporation will at some point find it necessary to raise additional equity to prevent the debt ratio from becoming too large. Thus, when we want to fix debt ratio we will change the ratio about equity that is the concept of weight average cost of capital. The following is a compute of example (Brigham, 1996).

WACC-[W.sub.d][K.sub.d] (1-T] + [W.sub.ps] [K.sub.ps] + [W.sub.s] [K.sub.s] =0.45(10%)(0.6)+0.02(10.3%)+0.53(13.4%) =10%

The following equation (3) is the fuzziness character of weighted average cost of capital to calculate fuzzy weighted average cost of capital. And the figure 2 that we change it into to fuzziness character model of fuzzy weighted average cost of capital. The [FWACC.sub.1] and [FWACC.sub.2] are not isosceles triangles, because they have the lowest and highest boundary, for example, [FWACC.sub.1] = (2, 2, 9) or [FWACC.sub.2]=(7, 10, 10). Thus, we will be known that left or right number and middle number are the same.

[FIGURE 2 OMITTED]

FWACC = [W.sub.d] [??] F[K.sub.d] [??] (1 [??] FT) [??] [W.sub.s] [F[K.sub.s] (3)

FWACC: Fuzzy Weighted Average Cost of Capital

[W.sub.d]: Debt Ratio

[W.sub.s]: Common Equity Ratio

F[K.sub.d]: Fuzzy Cost of Debt

F[K.sub.s]: Fuzzy Cost of Common Equity

FT: Fuzzy Margin Tax

3.6 Fuzzy Modified Internal Rate of Return (FMIRR)

We can modify the IRR and make it a better indicator of relative profitability, hence better for use in capital budgeting. The new measure is called the modified IRR or MIRR. Conclusion is that the modified IRR is superior to the regular IRR as an indicator of a project's "true" rate of return, or "expected long-term rate of return". Here COF refers to cash out flows (negative numbers), or the cost of the project, and CIF refers to cash in flows (all positive numbers). The left term is simply the present value (PV) of the investment outlays when discounted at the cost of capital, and the numerator of the right term is the future value of the in flows, assuming that the cash in flows are reinvested at the cost of capital. The future value of the cash inflows is also called the terminal value, or terminal value (TV). The discount rate that forces the PV of the TV to equal the PV of the costs is defined as MIRR (Brigham, 1996).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

C(PV): Investment Cost (Present Value)

COF: Cash out Flow at period t

K: Cost of Capital

The following equation (4) is the fuzziness character of Modified Internal Rate of Return to calculate fuzzy present value.

FC = [FTV/[(1 [direct sum]FMIRR).sup.n]] = [[n.summation over (t=0)][FCIF.sub.t][cross product][(1[direct sum]FWACC).sup.n-t]/[(1[direct sum]FMIRR).sup.n]]

FC: Fuzzy Present Value

FTV: Fuzzy Terminal Value

FCIF: Fuzzy Cash in Flow at period t

FWACC: Fuzzy Weighted Average Cost of Capital

FMIRR: Fuzzy Modified Internal Rate of Return

4. CASE STUDY OF BOT

4.1 Date of Case Study

This case is an airport BOT. According to Taiwan's law of incentive in investment on BOT. The corporation has minimum of 30% equity capital, other 70% capital can borrow from banks. It depends on what kind of capital to be constituted is good for corporation. In this case that corporation spend 4 years to build and 20 years to operate and income cash flow. The interest has to depend on the Taiwan's law of incentive in investment on Taiwan High Speed Rail BOT plan that borrow from many banks is between 3% and 5% (1999). The following table 1 is information about build fuzzy investment.

On the other hand, debt ratio ([W.sub.d]) is from 0%, 10%, 20%, 30%, 40%, 50%, 60% and 70%. The equity ratio is [W.sub.s] = 1 - [W.sub.d]. Fuzzy cost of common equity is [FK.sub.s]. The [FK.sub.s] is the cost including issue cost, agency cost and so on. The fuzzy tax (T) is (40%, 40%, 40%). The information is following as table 1 and 2.

4.2 Fuzzy Weighted Average Cost of Capital Computation

We use above BOT plan information to compute that before tax the fuzzy weighted cost of debt and fuzzy weighted cost of common equity result as following table 3. And fuzzy weighted average cost of capital to be used by equation (3), under difference [alpha]-cut is 0, 0.2, 0.4, 0.6, 0.8 and 1, as following table 4.

We have got [alpha]-cut of fuzzy weighted average cost of capital under difference debt ratio. Thus, according table 5 and equations (1) and (2), we can calculate left and right integration, [I.sup.L](FWACC) and [I.sup.R](FWACC), besides, addition [I.sup.L](FWACC) and [I.sup.R](FWACC) will get R(FWACC) numbers.

4.3 Capital Budgeting of Fuzzy Modified Internal Rate of Return

After fuzzy weighted average cost of capital calculation to be accomplished than under fuzzy weighted average cost of capital of range minimum number to compute fuzzy modified internal rate of return. We got the range minimum number R(FWACC)=5.394) that is when the debt ratio is 40%. Its fuzzy cost is (2.34%, 2.664%, 3.12%), thus, we use equation (4) and BOT plan of table 1, 2 to be calculated the FMIRR = (6.19%, 6.5%, 10.62%). It has explained very clearly. When fuzzy weighted average cost of capital_subtract from FMIRR and they have (3.07%, 3.836%, 8.28%), the benefit interval between 3.07% and 8.28%.

5. CONCLUSION

Under the difficult handle situation of cost of capital and capital budgeting. We need more information gained for finance planning management. But decision-makers usually are often uncertain and fuzziness on their knowledge. Under this situation, it is difficult to make good decision. Especially, the BOT investment not only to be developed by corporation but also has opposite benefit to government. About cost, income and risk of BOT that this paper construct of solution investment of BOT in finance. Besides, we create the WACC and MIRR methods on fuzzy concept. Based on this, we get an advantage of the fuzzy finance. The results of research explained under uncertain environment that decision marker more understand what is the interval of profit. In this case, we got the lowest at 2.34% interest, the higher at 3.12% interest. The investor will get what kind of cost they have to pay. On the other hand, the net benefit interval between 3.07% and 8.28%. The normal return level is at 3.836%. The investor also can get income and risk information from this methods. Thus, this model is offer a new thinking way and information supply for corporations.

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Kang-Lin Chiang, National Taiwan Ocean University, TAIWAN Kuang Lin, National Taiwan Ocean University, TAIWAN Hsuan-Shih Lee, National Taiwan Ocean University, TAIWAN Gin-Shuh Liang, National Taiwan Ocean University, TAIWAN

Dr. Kang-Lin Chiang earned his Ph.D. at National Taiwan Ocean University in department of shipping and transportation management, Taiwan.

Dr. Kuang Lin currently he is a professor of department of shipping and transportation management at National Taiwan Ocean University, Taiwan.

Dr. Hsuan-Shih Lee currently he is a professor of department of shipping and transportation management at National Taiwan Ocean University, Taiwan.

Dr. Gin-Shuh Liang currently he is a professor of department of shipping and transportation management at National Taiwan Ocean University, Taiwan.

TABLE 1. UNDER DIFFERENCE DEBT RATIO THAT FUZZY COST OF
CAPITAL, FUZZY COST OF COMMON EQUITY AND FUZZY TAX

[W.sub.d]     [FK.sub.d] (%)       [FK.sub.s] (%)         FT (%)

0%                  --             (2.7, 2.7, 3.2)     (40, 40, 40)
10%             (3, 3, 5)         (2.7, 2.72, 3.2)     (40, 40, 40)
20%            (3, 3.25, 5)       (2.7, 2.75, 3.2)     (40, 40, 40)
30%            (3, 3.5, 5)        (2.7, 2.84, 3.2)     (40, 40, 40)
40%            (3, 3.75, 5)       (2.7, 2.94, 3.2)     (40, 40, 40)
50%             (3, 4, 5)         (2.7, 3.05, 3.2)     (40, 40, 40)
60%            (3, 4.5, 5)        (2.7, 3.15, 3.2)     (40, 40, 40)
70%             (3, 5, 5)          (2.7, 3.2, 3.2)     (40, 40, 40)

TABLE 2. CASH IN FLOW OF OPERATE

Years         Cash in flows

1              (65, 72, 77)
2              (72, 77, 82)
3              (69, 74, 79)
4            (118, 123, 128)
5            (113, 118, 123)
6            (108, 113, 118)
7              (88, 93, 98)
8            (168, 173, 178)
9            (161, 166, 171)
10           (154, 159, 164)
11           (146, 151, 156)
12           (9160, 165, 170)
13           (152, 157, 162)
14           (144, 149, 154)
15           (135, 140, 145)
16           (127, 132, 137)
17           (140, 145, 150)
18           (131, 136, 141)
19           (9121, 126, 131)
20           (119, 124, 129)

Unit: million

TABLE 3. THE FUZZY WEIGHTED COST OF DEBT
AND FUZZY WEIGHTED COST OF COMMON EQUITY

[W.sub.d]       [W.sub.d]             [W.sub.s]
             [cross product]       [cross product]
            [FK.sub.d](%)(1-T)      [FK.sub.s](%)

0%                  --             (2.7, 2.7, 3.2)
10%         (0.18, 0.18, 0.3)    (2.43, 2.448, 2.88)
20%         (0.36, 0.39, 0.6)     (2.16, 2.2, 2.56)
30%         (0.54, 0.63, 0.9)    (1.89, 1.988, 2.24)
40%          (0.72, 0.9, 1.2)    (1.62, 1.764, 1.92)
50%          (0.9, 1.2, 1.5)     (1.35, 1.525, 1.6)
60%         (1.08, 1.62, 1.8)    (1.08, 1.26, 1.28)
70%          (1.26, 2.1, 2.1)    (0.825, 0.96, 0.96)

TABLE 4. FUZZY WEIGHTED AVERAGE COST OF
CAPITAL UNDER A=O, 0.2, 0.4, 0.6, 0.8 AND 1

Debt ratio     [alpha]       Fuzzy Weighted Average
                           Cost of Capital (FWACC) (%)

0%                0                [2.7, 3.2]
                 0.2               [2.7, 3.1]
                 0.4               [2.7, 3.0]
                 0.6               [2.7, 2.9]
                 0.8               [2.7, 2.8]
                  1                [2.7, 2.7]

10%               0               [2.61, 3.18]
                 0.2            [2.6136, 3.0696]
                 0.4            [2.6172, 2.9592]
                 0.6            [2.6208, 2.8488]
                 0.8            [2.6244, 2.7384]
                  1              [2.628, 2.628]

20%               0               [2.52, 3.16]
                 0.2             [2.534, 3.046]
                 0.4             [2.548, 2.932]
                 0.6             [2.562, 2.818]
                 0.8             [2.576, 2.704]
                  1               [2.59, 2.59]

30%               0               [2.43, 3.14]
                 0.2            [2.4676, 3.3056]
                 0.4            [2.5052, 2.9312]
                 0.6            [2.5428, 2.8268]
                 0.8            [2.5804, 2.7224]
                  1              [2.618, 2.618]

40%               0               [2.34, 3.12]
                 0.2            [2.4048, 3.0288]
                 0.4            [2.4696, 2.9376]
                 0.6            [2.5344, 2.8464]
                 0.8            [2.5992, 2.7552]
                  1              [2.664, 2.664]

50%               0                [2.5, 3.1]
                 0.2             [2.345, 3.025]
                 0.4              [2.44, 2.95]
                 0.6             [2.535, 2.875]
                 0.8               [2.63, 2.8]
                  1              [2.725, 2.725]

60%               0               [2.16, 3.08]
                 0.2              [2.304, 3.04]
                 0.4              [2.448, 3.0]
                 0.6              [2.592, 2.96]
                 0.8              [2.736, 2.92]
                  1               [2.88, 2.88]

70%               0               [2.085, 3.06]
                 0.2              [2.28, 3.06]
                 0.4              [2.475, 3.06]
                 0.6              [2.67, 3.06]
                 0.8              [2.865, 3.06]
                  1               [3.06, 3.06]

TABLE 5. RANGE OF FUZZY WEIGHTED AVERAGE COST OF CAPITAL

Debt ratio                           FWACC

             [I.sup.L](FWACC)   [I.sup.R](FWACC)   R(FWACC)

0%                 2.7                2.95           5.65
10%               2.619              2.904          5.523
20%               2.555              2.875           5.43
30%               2.524              2.879          5.403
40%               2.502              2.892          5.394
50%               2.4875             2.9125          5.4
60%                2.52               2.98           5.5
70%               2.5725              3.06          5.6325