On Fuzzy Internal Rate of Return

On Fuzzy Internal Rate of Return Christer Carlsson Institute for Advanced Management Systems Research, e-mail:christer.carlsson@abo.fi Robert Fullér Turku Centre for Computer Science, e-mail: rfuller@ra.abo.fi TUCS Turku Centre for Computer Science TUCS Technical Report No 211 November 1998 ISBN 952-12-0310-3 ISSN 1239-1891

Abstract We consider the internal rate of return IRR) decision rule in capital budgeting problems with fuzzy cash flows. The possibility distribution of the IRR at any r 0, is defined to be the degree of possibility that the fuzzy) net present value of the project with discount factor r equals to zero. We show that the possibility distribution of the IRR is a highly nonlinear function of quasi-triangular form. Furthermore we prove that the fuzzy IRR is stable under small changes in the membership functions of future cash flows. Keywords: Capital budgeting problem, internal rate of return, possibility distribution, sensitivity analysis TUCS Research Group Institute for Advanced Management Systems Research

1 Introduction Many decision making problems concern projects in which the cost and benefits accrue over a number of years. In this paper we consider only cases in which the costs and benefits are entirely monetary, such as the capital budegeting or capital investment decisions arising in commerce and industry. Authors consider two kinds of decision problems in capital budgeting: accept-or-reject and ranking. In accept-or-reject decisions, each project is considered independently of all other projects. Thus a portfolio of accepted projects is built up from several independent decisions. In ranking decisions, all the available projects are compared and ranked in order of favourability with the intention of adopting a single project: the most favourable. It should be noted that it is often important to include a null project representing the status quo; all the projects may be unfavourable compared with the alternative of adopting none of them if this is possible). Several decision rules have been suggested [1, 6, 9] to help decision makers rank projects which involve timestreams of costs and benefits, such as the payback period, accounting rate of return ARR), internal rate of return IRR) and net present value NPV). We shall briefly describe just the IRR decision rule. Let {a 0, a 1,..., a n } be a given net cash flow of a project a over n periods. We assume that a 0 < 0 as the project starts with an initial investment. The IRR, denoted by r, is defined to be the value of r such that the NPV of the project is zero. Thus find the IRR of a we need to solve Sa, r) := a 0 + a 1 1 + r + + a n 1 + r) n = 0 1) It is well-known that, if there is reinvestment in a project a i < 0 for some i 1) then its IRR may become ill-defined, i.e. equation 1) may have more than one solution. If the IRR of a project is ill-defined, it is not a suitable criterion to use in either accept-or-reject or ranking decisions. Suppose, however, that no project considered involves any reinvestment. Figure 1: Graph of a 2-year project a = 1,1,1) with IRR = 61.8%. Then NPV is a strictly monotone decreasing function of r and the equation 1) has a unique solution, moreover, the discount rate r can be interpreted in strictly financial terms as an interest rate. Now in an accept-or-reject decision it is clear that, if the market rate of interest is r 0, the project should be accepted if r > r 0 because this implies the that NPV at r 0 is positive. In comparing two projects, the one with the higher IRR should be preferred. 1

2 IRR with fuzzy cash flows More often than not future cash flows and interest rates) are not known exactly, and we have to work with their estimations, such as around 5, 000 in the next few years or close to 3 % ). Fuzzy numbers appear to be an adequate tool to represent imprecisely given cash flows [3, 4, 7, 13, 14]. Definition 2.1 A fuzzy number s a fuzzy set of the real line with a normal, fuzzy) convex and continuous membership function of bounded support. The family of fuzzy numbers will be denoted by F. A fuzzy number with a unique maximizing element is called a quasi-triangular fuzzy number. A fuzzy set s called a symmetric triangular fuzzy number with center a and width α > 0 if its membership function has the following form a t 1 if a t α At) = α and we use the notation A = a, α). If α = 0 then A collapses to the characteristic function of {a} IR and we write A = ā. A triangular fuzzy number with center a may be seen as a fuzzy quantity x is approximately equal to a. 1 a-α a a+α Figure 2: Symmetric triangular fuzzy number with center a. The use of fuzzy sets provides a basis for a systematic way for the manipulation of vague and imprecise concepts. In particular, we can employ fuzzy sets to represent linguistic variables. A linguistic variable can be regarded either as a variable whose value is a fuzzy number or as a variable whose values are defined in linguistic terms. Definition 2.2 A linguistic variable is characterized by a quintuple x, Tx), U, G, M) in which x is the name of variable; Tx) is the term set of x, that is, the set of names of linguistic values of x with each value being a fuzzy number defined on U; G is a syntactic rule for generating the names of values of x; and M is a semantic rule for associating with each value its meaning. 2

For example, if growth of the stock market) is interpreted as a linguistic variable, then its term set T growth) could be T = {slow, moderate, fast, very slow, more or less fast, sligthly slow,... } where each term in T growth) is characterized by a fuzzy set in a universe of discourse U = [0,100]. We might interpret slow as a growth below about 20 % p.a. moderate as a growth close to 60 % p.a. fast as a growth above about 100 % p.a. These terms can be characterized as fuzzy sets with membership functions 1 if v 20 slowv) = 1 v 20)/40 if 20 v 60 mediumv) = fastv) = { 1 v 60 /40 if 20 v 100 1 if v 100 1 100 v)/40 if 60 v 100 We will use triangular fuzzy numbers to represent the values of the linguistic variable [16] cash. If A = a, α) and B = b, β) are fuzzy numbers of symmetric triangular form and λ IR then A + B, A B and λa are defined by the extension principle in the usual way: A + B = a + b, α + β), A B = a b, α + β), λa = λa, λ α). Furthermore, if = a i, α i ) is a symmetric triangular fuzzy number and λ i = 1/1 + r) i, for i = 0,1,..., n, then we get A 0 + 1 + r) i = a 0 + a 1 1 + r + + a n 1 + r) n, α 0 + α 1 1 + r + + α ) n 1 + r) n. 2) Let A and B F be fuzzy numbers. The degree of possibility that the proposition s equal to B is true denoted by Pos[A = B] and defined by the extension principle as Pos[A = B] = sup min{ax), Bx)} = A B)0), 3) x IR The Hausdorff distance of A and B, denoted by DA, B), is defined by [12] DA, B) = max θ [0,1] max { a 1θ) b 1 θ), a 2 θ) b 2 θ) } 3

where [a 1 θ), a 2 θ)] and [b 1 θ), b 2 θ)] denote the θ-level sets of A and B, respectively. For example, if A = a, α) and B = b, α) are fuzzy numbers of symmetric triangular form with the same width α > 0 then DA, B) = a b. Lemma 2.1 [10] Let δ > 0 be a real number, and let A = a, α) and B = b, β) be symmetric triangular fuzzy numbers. Then from the inequality DA, B) δ it follows that { δ sup At) Bt) max t IR α, δ }. 4) β Let {A 0 = a 0, α 0 ), A 1 = a 1, α 1 ),..., A n = a n, α n )} be a given net fuzzy cash flow of a project A over n periods. By replacing the crisp cash flow values with fuzzy numbers in 1) we get A 0 + A 1 1 + r + + A n 1 + r) n = 0 5) where the equation is defined in possibilistic sense, and 0 denotes the characteristic function of zero. That is, the fuzzy solution [2] of 5) is computed by [ µ IRR r) = Pos A 0 + ] 1 + r) n = 0 = A 0 + ) 1 + r) n 0). for each r 0. Using the definition of possibility 3) and representation 2) we find Sa, r) 1 if Sa, r) Sα, r), µ IRR r) = Sα, r) 6) where we used the notations Sa, r) = a 0 + a 1 1 + r + + a n 1 + r) n, Sα, r) = α 0 + α 1 1 + r + + α n 1 + r) n We assume that a 0 < 0 the project starts with an initial investment), a 0 a 1 + +a n the project is at least as good as the null project), and a i 0, i = 1,..., n, no reinvestment). In this case we always get quasi-triangular fuzzy numbers for IRR in IR + 0 and equation 5) has a unique maximizing solution, r, such that, µ IRR r ) = max r 0 µ IRRr) = 1, and r coincides with r, which is the internal rate of return of the crisp) project a = a 0, a 1,..., a n ). Really, if r 0 then µ IRR r) = 1 if and only if Sa, r) = 0. As an example consider a 4-year project A = { 5, α), 3, α),4, α),6, α), 10, α)}, 4

with fuzzy IRR 6) where Sa, r) = 5 + 3 1 + r + 4 1 + r) 2 + 6 1 + r) 3 + 10 1 + r) 4 and Sα, r) = α 1 + 1 ) 1 + r + 1 1 + r) 2 + 1 1 + r) 3 + 1 1 + r) 4 It is easy to compute that µ IRR 0.781) = 1 for all α 0, so the maximizing solution to possibilistic equation 5) is independent of α. Figure 3: The graph of fuzzy IRR with a = 5,3,4,6,10) and α 0 = = α n = 0.5 Figure 4: The graph of fuzzy IRR with a = 5,3,4,6,10) and α 0 = = α n = 1 Figure 5: The graph of fuzzy IRR with a = 5,3,4,6,10) and α 0 = = α n = 5 However, as can be seen in Figs.3-5, the possibility distribution of the IRR is getting more and more unbalanced as the widths of the fuzzy numbers are growing. This means that when comparing the fuzzy IRR with the market interest rate r 0 in an accept-or-reject decision, the defuzzified value of µ IRR will 5

definitely differ from r whenever the process of defuzzification takes into account all points with positive membership degrees and not only the maximizing point). For example, all projects in Figs.3-5, have the same maximizing solution r = 0.781, but if we employ the center-of-gravity method then the defuzzified value of the project with α 0 = α 1 = = α n = 5 is around 0.84, which is esentially bigger in terms of rates of return) than 0.781. In ranking decisions we have to compare possibility distributions of a nonsymmetric quasi-triangular form. 3 Extensions We can also use non-symmetric triangular fuzzy numbers to represent the values of future cash flows. Definition 3.1 A fuzzy set s said to be a triangular fuzzy number with peak or center) a, left width α > 0 and right width β > 0 if its membership function has the following form At) = and we use the notation A = a, α, β). 1 a t α if a α t a 1 t a β if a t a + β 1 a-α a a+β Figure 6: Non-symmetrical triangular fuzzy number with center a. If A = a, α 1, β 1 ) and B = b, α 2, β 2 ) are fuzzy numbers of triangular form then A + B and A B are defined by: A + B = a + b, α 1 + α 2, β 1 + β 2 ), A B = a b, α 1 + β 2, α 2 + β 1 ), Furthermore, if = a i, α i, β i ), i = 0,1,..., n, are triangular fuzzy numbers then we get A 0 + 1 + r) i = a 0 + a 1 1 + r + + a n 1 + r) n, α 0 + α 1 1 + r + + α n 1 + r) n, β 0 + β 1 1 + r + + β n 1 + r) n ). 6

Let {A 0 = a 0, α 0, β 0 ), A 1 = a 1, α 1, β 1 ),..., A n = a n, α n, β n )} be a given net fuzzy cash flow of a project A over n periods. Then the fuzzy IRR of project s computed by Sa, r) 1 if Sa, r) Sα, r) 0 Sa, r), Sα, r) µ IRR r) = Sa, r) 1 + if Sa, r) 0 Sa, r) + Sβ, r) Sβ, r) In this case the shape of the fuzzy IRR is determined by the relationships between the left and right widhts of the fuzzy numbers in the project. Figure 7: The graph of fuzzy IRR with a = 5,3, 4,6) and α 0 = = α n = 1, β 0 = β 1 = = β n = 2 Figure 8: The graph of fuzzy IRR with a = 5,3, 4,6) and α 0 = = α n = 1, β 0 = β 1 = = β n = 3 Figure 9: The graph of fuzzy IRR with a = 5,3, 4,6) and α 0 = = α n = 5, β 0 = β 1 = = β n = 1 What if we allow trapezoidal fuzzy numbers, = a i, b i, α i, β i ), in a project? If the fuzzy numbers are not strictly unimodal then the set of maximizing solutions of the fuzzy IRR is a segment [r 1, r 2 ], where r 1 is the IRR of the project 7

a 0, a 1,..., a n ) and r 2 is the IRR of the project b 0, b 1,..., b n ). Therefore, if we employ the mean of maxima defuzzyfing method, then r 1 + r 2 2 is chosen for the most likely IRR value of the project. 4 Sensitivity analysis in fuzzy capital budgeting Consider two projects A = {A 0, A 1,..., A n } and A δ = {A δ 0, A δ 1,..., A δ n} with fuzzy cash flows = a i, α i ) and A δ i = aδ i, α i), i = 0,1,..., n. The fuzzy IRR of the project A δ, denoted by µ δ IRR, is computed by [ µ δ IRRr) = Pos A δ A δ ] i 0 + 1 + r) n = 0 = A δ A δ ) i 0 + 1 + r) n 0). for each r 0. Using the definition of possibility 3) and representation 2) we find µ δ 1 Saδ, r) if Sa δ, r) Sα, r), IRRr) = Sα, r) where we used the notation Sa δ, r) = a δ 0 + aδ 1 1 + r + + a δ n 1 + r) n. Let r δ) denote the IRR of the crisp project a δ = a δ 0, a δ 1,..., a δ n). That is, Sa δ, r δ)) = a δ 0 + a δ 1 1 + r δ) + + a δ n 1 + r δ)) n = 0 7) In the following we suppose that r δ) is the only solution to equation 7), i.e. a δ 0 < 0 and a δ i > 0 for i = 1,..., n. The next theorem shows that if the centers of fuzzy numbers and A δ i in projects A and A δ are close to each others, then there can only be a small deviation in the possibility distributions of their fuzzy IRR. Theorem 4.1 Let δ > 0 be a real number. If max{ a 0 a δ 0, a 1 a δ 1,..., a n a δ n } δ then { max µ IRRr) µ δ IRRr) min 1, r 0 δ α max }. 8) where α max = max{α 0, α 1,..., α n } µ IRR and µ δ IRR are the possibility distributions of IRR of projects A and Aδ, respectively. 8

Proof. It is sufficient to show that µ IRR r) µ δ IRRr) = [ ] Pos A 0 + 1 + r) n = 0 ) A 0 + 1 + r) n 0) [ Pos A δ 0 + A δ 0 + A δ ] i 1 + r) n = 0 = ) A δ i 1 + r) n 0) min { 1, } δ α 9) for any r 0, because 8) follows from 9). Using representation 2) and applying Lemma 2.1 to and A 0 + A δ 0 + we find D A 0 + 1 + r) n, Aδ 0 + a 0 + a 1 1 + r + + a n = Sa, r), Sα, r)), 1 + r) n A δ i 1 + r) n = Saδ, r), Sα, r)), 1 + r) n A δ ) i 1 + r) n = Sa, r) Sa δ, r) = a δ 0 + aδ 1 1 + r + + a δ n 1 + r) n ) a 0 a δ 0 + 1 1 + r a 1 a δ 1 1 + 1 + r) n a n a δ n n + 1) δ, for any r 0, and A 0 + sup A 0 + t IR ) 1 + r) n 0) A δ 0 + { } n + 1)δ max α 0 + α 1 + + α n ) 1 + r) n t) A δ 0 + { max { δ α max A δ ) i 1 + r) n 0) A δ ) i 1 + r) n t) n + 1)δ n + 1) max{α 0, α 1,..., α n } }. } = Which ends the proof. Theorem 4.1 can also be extended to fuzzy cash flows with arbitrary continuous) fuzzy numbers. 9

Theorem 4.2 Let δ > 0 be a real number. If then max{da 0, A δ 0), DA 1, A δ 1),..., DA n, A δ n)} δ max r 0 µ IRRr) µ δ IRRr) min{1, ωδ)}. where ωδ) denotes the maximum of moduli of continuity of all the fuzzy numbers in projects A and A δ at point δ. The proof of this theorem is carried out analogously to the proof of Theorem 3.1 in [8]. 5 Concluding remarks In this paper we have shown that the fuzzy IRR has a stability property under small changes in the membership functions representing the fuzzy cash flows. Nevertheless, the behavior of the maximizing solution, r δ), of possibilistic equation A δ A δ i 0 + 1 + r) n = 0, towards small perturbations in the membership functions of the fuzzy coefficients can be very fortuitous. That is, the distance r r δ), which coincides with r r δ), the distance between the internal rates of returns of crisp projects a = a 0, a 1,..., a n ) and a δ = a δ 0, a δ 1,..., a δ n) if = a i, α i ) and A δ i = aδ i, α i), i = 0, 1,..., n) can be very large even for very small δ. In this manner, the fuzzy model can be considered a well-posed extension [11, 15] of the generally) ill-posed crisp internal rate of return decision rule. References [1] J. Baron, Thinking and Deciding, 2nd edition, Cambridge University Press, 1994). [2] R.E. Bellman and L.A. Zadeh, Decision-making in a fuzzy environment, Management Sciences, Ser. B 171970) 141-164. [3] J.J. Buckley, Portfolio analysis using possibility distributions, in: E. Sanchez and L.A. Zadeh eds., Approximate Reasoning in Intelligent Systems, Decision and Control, Pergamon Press, New-York, 1987 69-76. [4] J.J. Buckley, Fuzzy mathematics of finance, Fuzzy Sets and Systems, 211987) 257-273. [5] C. Carlsson and R. Fullér, On fuzzy capital budgeting problem, in: Proceedings of the International ICSC Symposium on Soft Computing in Financial Markets, Rochester, New York, USA, June 22-25, 1999, ICSC Academic Press, 1999 submitted). 10

[6] Robert T. Clemen, Making Hard Decisions: An Introduction to Decision Analysis, 2nd edition, Duxbury Press, An Imprint of Wadsworth Publishing Company, Belmont, California, 1996). [7] C. Chiu and C.S. Park, Fuzzy cash flow analysis using present worth criterion, The Engineering Economist, 391994) 113-138. [8] M.Fedrizzi and R. Fullér, Stability in possibilistic linear programming problems with continuous fuzzy number parameters, Fuzzy Sets and Systems, 471992) 187-191. [9] Simon French, Readings in Decision Analysis, Chapman and Hall, London, 1990). [10] R.Fullér, On stability in possibilistic linear equality systems with Lipschitzian fuzzy numbers, Fuzzy Sets and Systems, 341990) 347-353. [11] R. Fullér, Well-posed fuzzy extensions of ill-posed linear equality systems, Fuzzy Systems and Mathematics, 51991) 43-48. [12] R.Goetschel and W.Voxman, Topological properties of fuzzy numbers, Fuzzy Sets and Systems, 101983), 87-99. [13] E.E. Haven, The use of fuzzy multicriteria analysis in evaluating projects with identical net present values and risk, in: M. Mares et al eds., Proceedings of Seventh IFSA World Congress, June 25-29, 1997, Prague, Academia Prague, Vol. III, 1997 154-157. [14] C. Kahraman and Z. Ulukan, Fuzzy cash flows under inflation, in: M. Mares et al eds., Proceedings of Seventh IFSA World Congress, June 25-29, 1997, Prague, Academia Prague, Vol. IV, 1997 104-108. [15] M. Kovács, A stable embedding of ill-posed linear systems into fuzzy systems, Fuzzy Sets and Systems, 451992) 305 312. [16] L.A.Zadeh, The concept of linguistic variable and its applications to approximate reasoning, Parts I,II,III, Information Sciences, 81975) 199-251; 81975) 301-357; 91975) 43-80. 11

Turku Centre for Computer Science Lemminkäisenkatu 14 FIN-20520 Turku Finland http://www.tucs.abo.fi University of Turku Department of Mathematical Sciences Åbo Akademi University Department of Computer Science Institute for Advanced Management Systems Research Turku School of Economics and Business Administration Institute of Information Systems Science