A POSSIBILISTIC APPROACH TO CALCULATING THE INTERNAL RATE OF RETURN FOR INVESTMENT PROJECTS WITH FUZZY CASH FLOWS

Abstract

The internal rate of return (IRR) is among the most widely used characteristics of investment projects. In the deterministic case the IRR can be found as a solution to an algebraic equation. The existence and applicability of the IRR may be guaranteed by rather general conditions, e. g. those of Norstrom theorem. Under uncertainty, the IRR ceases to be uniquely determined even for typical projects (a project is typical if costs precede returns). Under uncertainty, the IRR can take values from a more or less large set of possible values. A key problem is to transform the information about initial parameters into a distribution of the IRR values. Probabilistic methods can be used if quite a number of stringent requirements are met. If an unusual or extraordinary project is considered the probabilistic approach is not sufficiently substantiated. In such cases possibilistic methods and fuzzy set theory seem to be more suitable. The present paper aims to provide a method for evaluating the IRR of investment projects with fuzzy cash flows using the possibility theory. Given a fuzzy cash flow, the IRR is presented as a fuzzy set and the membership function may be considered as a version of distribution. Under the standard addition of fuzzy numbers we give explicit formulas for the membership function of the IRR. If components of a fuzzy cash flow are correlated we use the addition of fuzzy numbers with respect to t-norms. Generally, a possibilistic evaluation of the IRR with respect to a non-standard t-norm is rather difficult and was not considered before. If the t-norm is generated by a convex additive generator we reduce the evaluation of the IRR to a common convex optimization problem. A numerical example is presented. We believe the proposed method can be applied to evaluating the efficiency