Guerra M. L.

MAGNI, Carlo Alberto

Stefanini L.

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Average internal rate of returnwith interval arithmetic∗Maria Letizia Guerra† Carlo Alberto Magni‡Luciano Stefanini§1 Extended abstractThe internal-rate-of-return investment choice method is widely used to evaluateand compare projects; the decision is based on the ranking of the Internal Rateof Return (IRR) projects and finally on the choice of the higher one. Manycontributions in the last eight decades have been devoted to the corrections ofthe method that can reserve many technical difficulties, misunderstandings andincompatibility with the net-present-value rule.The use of a rate of return for investment decisions under uncer-tainty can be approached in several ways. The debate about the use of thetheory of probability to model uncertainty in economics has a long history andin investment decisions many weak points have been focused.The interval and the fuzzy arithmetics represent alternative ways to modeluncertainty in investment decisions: here we summarize the results achieved insome contributions.The studies in [7] go into the direction of studying the possible ways tomodel uncertainty in economics; in particular, they show that robustness of cap-ital budgeting techniques depends on the statistical estimation of cash amountsand interest rates, exhibiting often hard difficulties that can be captured by fuzzynumbers. In [12] methods for integrating probability and possibility distribu-tions are discussed and a computer simulation used for an investment projectrisk assessment shows that it is possible to model some parameters with fuzzynumbers and others with a probability distribution.Kuchta in [8] proposes generalized fuzzy equivalents in order to incorpo-rate uncertainty into the most commonly used techniques for capital budgeting.These fuzzy equivalents allow to evaluate projects whose cash flows and/or du-ration are not known precisely, but given in the form of fuzzy numbers. In[4] we follow the Kuchta approach with particular attention to the parametric∗This research was partially supported by the National Project PRIN (2008JN-WWBP 004): Models and Fuzzy Calculus for Economic and Financial Decisions, financedby the Italian Ministry of University.†Department Matemates, University of Bologna, Italy‡Department of Economics, University of Modena and Reggio Emilia, Italy§Department of Economics, Society and Politics, University of Urbino ”Carlo Bo”, Italy1representation of fuzzy numbers. In a second paper (details in [9]) Kuchta as-sumes a fuzzy Net Present Value for some projects and shows that the commonrealization of them may allow savings in the resource utilization.In [1] the concept of possibilistic mean and variance is extended to adaptivefuzzy numbers and applied to the computation of the fuzzy net present value offuture cash flows.Also in [6] the underlying hypothesis is that fuzzy variables can reflect un-certainty of investment outlays, annual net cash flows and investment capital; anew mean-variance model based on credibility measure is proposed for optimalcapital allocation.The potentialities of the fuzzy approach are highlighted also in [13], fewerassumptions about the data distribution and market behavior are enough to de-scribe fluctuations that go beyond the probability model. A measure of the riskassociated with each investment opportunity and an estimate of the projects’robustness towards market uncertainty is then derived.A fuzzy logic system in [15] is adopted to extend the classical discounted cashflow model in order to take into account the uncertain information intrinsic inthe value of a company’s financial asset.In [10] a fuzzy binomial approach for project valuation under uncertaintyis proposed and a method to compute the mean value of a project’s fuzzy NetPresent Value is provided.The approach we choose in order to manage uncertainty in investment deci-sion making is the definition of variables as intervals.The arithmetic of data represented with intervals is well established andformally consistent; in particular it is possible to manage the four operationsthrough the extension principle in order to build an arithmetic based on theextended operations.The key aspect is, in fact, the concept of equation that has to be definedin a rigorous way; some misunderstanding can be eventually produced by thesemilinearity structure of the intervals space. The analysis must be much carefulwhen dealing with the uncertainty propagation that is an implicit feature in theintervals arithmetic.In order to give an overview on the meaning of the uncertainty in intervalsarithmetic we can think about the fact that the length of the sum of two intervalsis equal to the sum of each interval itself. So the equation A + X = B admitssolution only when the length of A is not bigger than the length of B and it isnot equivalent to A = B−X. The non equivalence holds in terms of the way inwhich the uncertainty propagates: the position of X at the left or at the rightchanges the propagation.In order to deeper analyze the true meaning of the equation A+X = B whenthe variables are intervals, we have to distinguish between the input data andthe variables to be determined; it follows that the equation may have solutiononly in the arithmetic form A+X = B or A = B −X where the two equationsare equivalent if and only if A and B are real numbers, in other words onlywhen there is no uncertainty on the values.2The same reasoning can be applied to the multiplication and to more generalequations, involving two operations, like Ax+B = C.Any time we want to define a variable varying in an interval in terms ofother variables by mean of the arithmetic is like to define a variable X suchthat A + X = B and to interpret the left side A + X as it is the same as B.However to find such a variable X has a different meaning if we think about theinterval that has to be equal to B −AIn the intervals arithmetic it is possible to write:X = B 	gH Awhere 	gH is the generalized Hukuhara difference, whose properties areestablished in [14], defined as:X = B 	gH A⇔ A = B +XorB = A+ (−1)XThe crucial aspect requires that the applications based on the intervals arith-metic have to be studied contextually to an epistemic analysis on the meaningof the relations between variablesWhen managing with relations having a financial or economic nature, theuncertainty propagation becomes an urgent analysis.In this paper we follow the new perspective of the Average Internal Rateof Return (AIRR), introduced in [11], where the IRR is dismissed andreplaced by a mean of the project’s one-period rates, weighed by theoutstanding capitals invested in each period. We generalize [11] bymodeling the uncertainty of some key variables by means of intervals. Theinterval arithmetic and the interval ordering we apply are introduced in [5].The basic variables for a valuation project are now defined in their interval-valued form. The temporal sequence is expressed in the classical form as t ∈{0, 1, 2, ..., T} , the cash flow at time t is defined as xt =[x−t , x+t], the interestrate is r = [r−, r+] and the discount rate is v = [1 + r−, 1 + r+] = [v−, v+] underthe condition 1 + r− > 0. Given the above definitions, the following identity isverified: (11 + r)t=[1(1 + r+)t ,1(1 + r−)t]=[1(v+)t ,1(v−)t]and the Net Present Value (NPV) of the project, with values varying in aninterval, isPV (x | r) =T∑t=0xt (1 + r)−t=[PV − (x | r) , PV + (x | r)]where, in order to obtain PV − (x | r) =T∑t=0x−t(1+r+)tand PV + (x | r) =T∑t=0x+t(1+r−)t ,3the use of the extension principle is recommended in a rigorous way, producing:PV (x | r) =minξt∈xts∈rT∑t=0ξt(1 + s)t , maxξt∈xts∈rT∑t=0ξt(1 + s)tThe Net Future Value (NFV) is the future value of NPV and is defined as:PVt (x | r) = (1 + r)t PV (x | r) t ≥ 1.The NPV rule rule implies that a project is worth undertaking if andonly if PV (x | r) > 0 that is equivalent to PVt (x | r) for every t. The internalrate of return (IRR) of the project is a value k that satisfies: PVT (x | k) = 0.However, the latter is never defined in the interval case. By contrast,the AIRR is well defined even in the interval case and consistent withthe (interval) NPV.Definition 1 Let ct =[c−t , c+t]be the interval-valued invested capital (or bor-rowed capital) in period [t− 1, t] t = 0, 1, ..., T. Assume c0 = −x0 =[−x+0 ,−x−0 ](also assume, by definition, cT = 0 = [0, 0] . The return Rt for period [t, t+ 1]is defined to satisfy, formally,:ct−1 +Rt = ct + xt (1)that is equivalent to:1. ct−1 = (ct + xt)	gH Rt2. Rt = (ct + xt)	gH ct−13. ct = (ct−1 +Rt)	gH xt4. xt = (ct−1 +Rt)	gH ctdepending on the value that requires to be computed and on the values that areknown (with the corresponding uncertainties).The uncertainty propagation produced by (1) is the key feature of the fuzzyversion of the new AIRR model; from this more general perspectivean interval average rate of return may be derived, which enablesinvestors to deal with uncertainty in a rather sophisticated way.References[1] S. S. Appadoo, S. K. Bhatt, C. R. Bector, Application of possibility theoryto investment decisions, Fuzzy Optim Decis Making, 7 (2008) 35–57.[2] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications,Academic Press, New York, 1980.[3] D. Dubois, H. Prade (Eds.), Fundamentals of Fuzzy Sets, The Handbooksof Fuzzy Sets Series, Kluwer, Boston, 2000.4[4] M.L. Guerra, L. Sorini, L. Stefanini, Fuzzy Investment Decision Making,L.Magdalena, M. Ojeda-Aciego, J.L. Verdegay (eds.): Proceedings of IPMU08, ISBN: 978-84-612-3061-7, (2008), 745-750.[5] M.L. Guerra, L. Stefanini, A comparison index for interval ordering, 2011IEEE Symposium on Foundations of Computational Intelligence (FOCI2011) Proceedings, IEEE Catalog Number: CFP11FOC-CDR, ISBN: 978-1-4244-9980-9, 53-58.[6] X. Huang, Mean-variance model for fuzzy capital budgeting, Computers &Industrial Engineering 55 (2008) 34–47.[7] C. Kahraman, D. Ruan, E. Tolga, Capital budgeting techniques using dis-counted fuzzy versus probabilistic cash flows, Information Sciences, 142(2002) 57–76.[8] D. Kuchta, Fuzzy Capital Budgeting, Fuzzy Sets and Systems, 111 (2000)367-385.[9] D. Kuchta, A fuzzy model for R&D project selection with benefit, outcomeand resource interactions, The Engineering Economist, 46, (2001) 164–180.[10] S. H. Liao, S.H. Ho, Investment project valuation based on a fuzzy binomialapproach, Information Sciences, 180 (2010), 2124-2133.[11] C.A. Magni, Average internal rate of return and investment decisions: anew perspective, The Engineering Economist, 55 (2010) 150-180.[12] B. Rebiasz, Fuzziness and randomness in investment project risk appraisal,Computers & Operations Research, 34 (2007) 199–210.[13] A. Serguieva, J. Hunter, Fuzzy Interval Methods in Investment Risk Ap-praisal, Fuzzy Sets and Systems, 142 (2004) 443-466.[14] L. Stefanini, A generalization of Hukuhara difference and division for inter-val and fuzzy arithmetic, Fuzzy Sets and Systems, 161, 11 (2010)1564-1584.[15] J. S. Yao, M. S. Chen, H. W. Lin, Valuation by using a fuzzy discountedcash flow model, Expert Systems with Applications 28 (2005) 209–222.[16] L. A. Zadeh, Fuzzy Sets, Information and Control, 8 (1965) 338-353.5

Average internal rate of return with internval arithmetic

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