Computing the Internal Rate of Return (includes associated paper 15843)
Authors: Wolff Martin , Van Rensburg W.V.J. 
Published: December 2007

Summary
Computing the internal rate of return(IRR) of an investmentefficiently and under a wide range of conditions is a problem ofteninsufficiently addressed in finance literature. This paper reviewsarticles and books that deal with the IRR. The term is defined, andexamples are given. Problems that arise in computing and using the IRR arediscussed. Finally, a program is presented that computes and analyzes theIRR efficiently for a wide range of cash flows.

Introduction
The IRR is a widely used criterion for measuringinherent project acceptability and for comparing and rankingdifferent projects. The literature points out a number ofdefects, some of which remain controversial. This paperreviews some of the literature on the subject, particularly papers that deal with the IRR on a quantitative basis, and presents a computer program that calculates ordescribes the IRR under a wide range of conditions.

Definitions of the IRR
The simplest definition of the IRR. as stated by Jean, is the interest rate that makes the net present value (NPV)of a project equal to zero. If a curve of NPV vs. discountrate (DR) is drawn, the IRR ideally is the intersection ofthis curve with the x axis, which may never occur or may occur one or more times.
Several other definitions of the IRR exist. Cissell andCissell state that the IRR is the rate that makes theinflows and outflows equal at a certain point in time. Thisis essentially the same as the first definition because a zeroNPV implies zero value at all times.
Renwick has two definitions that help to clarify theway the IRR works. The IRR is the equivalent of therequired rate of interest on a savings account, with positivecash flows viewed as withdrawals and negative cash flowsviewed as deposits, so that the balance is zero at the endof the project. Alternatively, the IRR can be viewed asdenoting total profits expressed as a percent of totalinvestment outlay, as opposed to the NPV, which measurestotal dollars of net profit directly.
Bernhard defines the IRR with an equation. First, to define the initial investment (PO, which is usuallynegative) and succeeding cash flows (PI to P, for Years1 through n), the IRR is the interest state such that
..........................................(1)
He refers to this as the simple IRR. where a moregeneral IRR is the set of IRR's that solve the following equation:
..........................................(2)
This allows interest rates to vary from year to year as inreal life but produces a measure that is very difficult tocompute or to use. The simple IRR is a special case ofthe general one, where all the IRR's are equal. Someinteresting consequences of this approach are discussedlater.
Bernhard also points out that the literature usesmany alternative terms for the IRR, including yield, marginal efficiency of capital, profitability index, interest rateof return, and the project rate of return by the discounted-cash-flow, investor's, or scientific method.

Computation of the IRR
Literature on the actual means of computing the IRR isquite varied. A good portion of the literature states thatthe IRR is usually found by trial and error. Vichaspresents a technique based on interpolation of financial-table values that is valid but of limited accuracy. In fact, much better techniques are available.
Eq. 1 can be rewritten as a polynomial in x by makingthe substitution x = 1 / (1 + IRR) as follows:
..........................................(3)
Finding IRR = (1/x) - I corresponds to finding acceptablereal roots to Eq.
Some authors have restricted therange of acceptable IRR's to positive real numbers, making the range of x 0 is lesser than x less than 1. Other authorsallow consideration of negative IRR's down to 1 and work on the range 0 less than x less than oo. It is clear that projects with negative IRR's are not normally acceptable. but externalconsiderations and ranking requirements could mean suchprojects need to be considered.
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