Internal rate of return : an investment decision criterion of full applicability

applicability

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Hajdasinski, M. M. (1983). Internal rate of return : an investment decision criterion of full applicability. (EUT -BDK report. Dept. of Industrial Engineering and Management Science; Vol. 1). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1983

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University of Technology

the Netherlands

Department of

Industrial Engineering

&

Management Science

Internal Rate of Return - an Investment Decision Criterion of Full Applicability by

Mirosaw M. Hajdasinski

Report EUT /BDK/1 ISBN 90-67S7-001-X Eindhoven, 1983

by

Miroslaw M. Hajdasinski

Report EUT/BDK/1

ISBN 90-67S7-001-X

Eindhoven, 1983

Eindhoven University of Technology

Department of Industrial Engineering

&

Management Science

Eindhoven, Netherlands

MirosXaw M. Hajdasinski SUM MAR Y

One of the most commonly used decision criteria in investment evaluation is the Internal Rate of Return.

Although it is used so often in practice it is nevertheless criticised just as often by theorists and compared with the criterion of the Net Present Value, which the theorists con-sider to be better and more objective.

In the present study it is shown that the bases for criticism of the Internal Rate of Return are not the consequence of false premises on which this criterion is built up, but are the consequence of an incorrect procedure in handling of the criterion and its inconsistent and one-sided interpretation. If these faults are corrected and the criterion of the Internal Rate of Return is correctly interpreted in accordance with its real content, it provides the same results in identical decision situations as does the Net Present Value criterion and is, moreover, even more informative than the last-named. This proves that there is in reality no conflict between the two criteria, Internal Rate of Return and Net Present Value, nor can there be such a conflict, so that this can be consi-dered as the definite end to a dispute carried out with much vehemence and passion throughout the western hemisphere over a long series of years.

*

MirosXaw M. Hajdasinski

1. Introduction

A very large number of different assesment criteria are

known in the theory of investment calculation but only a few enjoy wider application.

Practice is in fact very conservative and only a small number of well-tried criteria providing the user with "cut-and-dried"

judgments which are not only the opposite of abstract but are also easy to apply, have enjoyed preference for a considerable number of years.

As has been shown from inquiries and studies carried out in a number of countries, [6]'[7] ,[11] ,[12] ,[14] ,[16] ,[18] ,[19J,

[2~, the two investment criteria most often used by the lea-ding international companies are Net Present Value (NPV) and Internal Rate of Return (IRR). With the aid of the criteria, far-reaching decisions whose degree of objectivity depends directly on their proper application, are made daily in the free market oriented world. It is therefore not surprising that these two criteria have, for years, been the subject, not only of great interest but also of considerable discussion in the professional journals concerned.

In this discussions, which took place parallel to and largely independently of one another in the English and German-speak-ing countries, the main issue is that, in identical situations the judgment by the NPV criterion on the one hand can easily be in conflict with that of the IRR criterion on the other, thus repeatedly placing researchers in the unenviable situ-ation of feeling obliged to decide as to which is preferable.

*

The Stanislaw Staszic University of Mining and Hetallurgy, Cracow, Poland

The aim of the present study is to show that there is not, and cannot be any such discreteness between the two criteria and that their judgements always lead conclusively to identical investment decisions.

2. Application of the two criteria 2.1 Investment alternatives

A problem as to a decision on an investment, of whatever kind, always arises when i t comes to choosing one out of a number

(at least two) of investment alternatives, each of which pre-cludes the other. Such an alternative can take the form of a single investment project or is just as likely to be several projects which do not exclude one another and whose terms can fallon different dates.

As each such investment project can be described, among other things by its Cash Flow Pattern (CFP) i t follows that when the CFP's of all the investment projects forming the said alternatives are algebraically summed up within the individu-al years of their term of duration, the totindividu-al CFP of the

alternative under consideration can be obtained. Thus, all the investment alternatives open to selection can be described by means of their CFP's.

The terms, not only of the individual projects, creating an alternative but also of any two alternatives, can differ from one another, depending on the diversity of tasks on which the intended investment ac.tivity is based.

If we now take one year as the unit of time and 0 as the earliest and n the latest year of all the terms of the in-dividual alternatives, we obtain a referent period of time

[o,n] which embraces the terms of all the investment natives. If thecash flows of the individual investment alter-natives in the years outside this referent period of time are taken as zero, i t is now possible to take any set of invest-ment alternatives having different terms and convert i t into

a set of alternatives with uniform terms, all of which would be identical with the combined period of time envi-saged. Thanks to this possibility we shall henceforward

consider only such mutually exclusive investment alternatives which have identical terms.

2.2 Definition of the two decision criteria

- - -

-Let us now take

N - as the number of mutually exclusive investment alter-natives open to selection,

a, _{J ,} k as the cash flow of the investment alternative k in

the year j ( j = 0, 1 , .•• ,n ; k = 1, 2 , ••. , N)

i - a s the constant interest rate in the whole period of time referring,

then, on the assumption that the cash flows always come at the end of the year in question, for the net present value NPV

k of the kth investment alternative, which we shall consider a function of the interest i , we obtain the fol-llowing formula:

n NPV

k

=

NPVk (i) =

L

_j=O aJ, ,k (1 + i) - j , i E (-1 ,+ o(» (1) This function is continuous and differentiable in (-1,+00)

with the following limits: lim NPVk(i) =

i -+ o<J

lim NPVk(i) = + oC .sgn(acf. k) ,

i~ -1 k'

(2 )

(3 )

where j = fk and j

=

d.

k

( t

k'

~

E { 0 ,1 , .•. , n} ) respec-tively - the first and last year,in each case,in which a_j ,k

#

°

applies wi thin the CFP

k of the k th investment alternative.

If we were confronted, for instance with n = 9 with

+300, +800, +1500,0,0,0}, then we should have

r

k

=

2 ,

~

=

6 and consequently: lim NPVk(i)

=

-100 and i---+ 00

lim NPV

k (i) = + 0<)

i - - 1

In connection with the continuity of function (1) for

i E (-1,+ ( 0 ) i t is found from (2) and (3) that in the event of differences in the sign of a and

a.r

(that is

rk,k Qk,k

when sgn (ark,koa!k,k) = -1) has at least one root in (-1,+ O()).

On the other hand, if i t is assumed that n

sgn(aV'" kO

L::

a j k)

=

-1 ,

ok' j =0 '

i t would mean that the function NPVk(i) has at least one positive root 0

(4 )

Cases are often found in practice, in particular when an

investment alternative is in fact a single investment project, that the function (1) has only one positive root r

k ' which we shall describe in the following considerations as Internal Rate of Return of the kth investment alternative. Its value is thus obtained from the equation:

(5 )

An investment alternative CFP can have no change in sign, one change, or several changes in sign, and in general con-siderations such as those dealt with here, none of the possibilities may be excluded.

However, i t is considered "classical" in investment calcu-lation to have only one change of sign where a series of negative cash flows is followed by a series of positive ones. The whole philosophy of investment is reflected in such a CFP, according to which a period of investment outlays is followed by a time of investment returns.

As has long been known, the Net Present Value function of such a classical CFP always has an IRR which, according to (4), can be positive when the inequality

n

La'k>o

j =0 J,

is fulfilled. In that case the curve of the NPV function will take one of the two forms shown in figure 1.

(6)

The diagram given in figure 1a is that of an NPV function whose CFP has only one negative cash flow. Figure 1b, on the other hand, illustrates the situation when the CFP con-cerned contains at least two negative cash flows. The NPV function will then always have a minimum.

A CFP with only one change of sign, but in reverse, would be just as "classical" as the CFP discussed in the forego-ing paragraph. It would show the creditforego-ing mechanism and the diagrams of its NPV function, likewise the conditions for the existence of the positive IRR are also immediately evident from the foregoing considerations, provided the appropriate changes in signe are made.

2.3 Selection of the best investment alternative in the

- - - -

-traditional manner

-The manner in which the most favourable investment alterna-tive is still selected in the majority of cases is descri-bed below:

(a) Each alternative is first of all checked as to its absolute profitability by taking the minimum accep-table return (MAR) m as the measure of the limiting profitability.

(b) All the mutually exclusive investment alternatives are compared to find the one which is, relatively speaking, the most profitable one among them. This procedure can, for both criteria, be described as follows:

Net Present Value Criterion

(a) The NPV's are obtained for all N investment alter-natives and for i

=

m and compared with 0 .

Let us assume that N1 ~ N alternatives, which we describe as k: ' s = _{1,2, ... ,N 1 ; k:e:{1,2, ... ,N},}

have fulfilled the condition

NPV_k (m)

>

0 ( 7)

(b) From the number of approved investment alternatives

*

N1 ' the best k is found, that is whose NPV satis-fies the relation

NPV_k

*

(m) • max [ NPV< (m)] s = 1,2, ••• ,N₁ (8)

Internal Rate of Return Criterion

(a) On the proviso that all the investment alternatives in the range

[0,

+ 0<:) ) have one and only one r

k ' all the IRR's r

k , k

=

1,2, ••• ,N are compared with the MAR m. The general pOint of departure is that all the CFP's begin with negative cash flows and that the coverage condition (6) is met, which means that the corresponding NPV functions have a curve within the range

[0,

+ 0<> ) as shown in figure

1.

Under these premises the criterion of absolute profita-bility is then:

( 9)

and we shall assume that this inequality is fulfilled by N2 ~N investments alternatives, which we describe

(b) Out of the number N2 of approved alternatives, the k+ th

best investment alternative is sought for, for s

s

=

1,2, ... ,N 2 is true.

3. Conflict between two decision criteria - criticism of internal rate of return

In practice the two decision criteria NPV and IRR often

(10)

lead to identical decisions, but i t can occur that two different alternatives are determined as the best in each case. In illustration let us take as an example two invest-ment alternatives A and B which are described for n

=

3 by the CFP's: CFP

A and CFPB, as shown in Table 1. TABLE 1 j 0 1 2 CFP A -2000 +900 +1010 CFP B -1000 +500 +440 CFP(A_B) -1000 +400 +570

Both alternatives have each a positive IRR (15%) and r

B

=

0.20 (20%) which, at the MAR 3 +690 +480 +210 r_A

=

0.15

m

=

0.08 (8%), means that on the IRR criterion every alternative is absolutely profitable and that alternative B is

super-ior to alternative A.

However, if we take the NPV criterion into consideration, we obtain the following values for m

=

0.08:

NPV_A(0.08)

=

246.99 and NPV_B(0.08)

=

221.23 which, though they confirm the absolute profitability of the two alternatives, nevertheless show that A is superior to B. This state of affairs is shown in figure 2, in which i t can be seen that the pOint of intersection of the two NPV curves lies at about 0.0954. This means that the judgments of the

criteria under consideration are conflicting so long as m lies in the range [0 ; 0.0954J. If, however, we were dea-ling with m> 0.0954 the two criteria would not be conflic-ting.

The fact that the agreement or disagreement in the message contained in the NPV and IRR criteria depends on the height of the assumed MAR, was the first thing that caused many critics to move into the attack, remarkably enough immedia-tely singling out the IRR criterion as the target for the onslaught, as if the mere fact that two decision criteria differed from one another in their effect was enough justi-fication for their judgment as to which of the two was right. A second reason for criticism of the IRR criterion was pro-vided by the fact that the NPV fUnction in the range (-1,+00)

can have either no root, or several roots, positive ones in particular, in which was seen a situation giving rise to partly insurmountable difficulties in interpretation.

A further grave objection to the IRR was the fact that this criterion takes no account at all of the absolute amount of money invested. For instance, this is because an alternative

in which an investment of Fl. 100 gives a return of 10% is held to be just as profitable as one of Fl. 1000 yielding the same rate of profit.

Although this does not exhaust the grounds on which the IRR was subjected to criticism

[3],

the three above-mentioned were far and away the most serious and have to date led to this criterion being regarded in many circles as being of no importance.

As the proponents of the Internal Fate of Return, on their part also expressed criticism of NPV, a situation came about calculated to arouse extreme disconcertion among investors. If one were to consider all the arguments on each side as well founded, one would, when making decisions, dispense completely with IRR as well as NPV. As they so often do, the

practical investors retained their sang froid, refused to be impressed by the squabbles around the incriminated criteria and went on making unswerving use of them [6J ,[7J ,[8],[11],

[12J

,[14J,[16J ,[18J , [1'9J ,[20J.

4. The causes of the misunderstandinss

All the disputes over the usefulness of the two criteria, wich went on for years on end, in particular with regard to the Internal Rate of Return have, in the opinion of the author,two main causes which are given below:

(1) incorrect procedure in the comparison of two mutually exclusive investment alternatives;

(2) incorrect interpretation of the IRR criterion.

4.1 fo!paris~n_o! two !u~u!l!y_e~c!u!i~e_i£v~s~m~n! !l!eEne-tives

If i t is desired to determine the relative profitability of two mutually exclusive alternatives A and B , the point of de-parture must be to measure the superiority in returns of alter-native A over those of B. As the performance of an invest-ment return, viewed from the economic standpoint, is charac-terised by its CFP, that means that the difference in yield on the part of alternative A compared to B will be character-ised by such a CFP obtained in the individual years from the difference in the cash flows arising ou~ of the two alternatives A and B.

Let us call this difference CFP(A_B) and consider i t hence-forth, for the sake of simplicity, as the CFP of a fictitious difference alternative (A-B) .

If we now describe the CFP's of alternatives A and B by CFP and

CFP

A = {aj,A} j = O,1, ••• ,n ( 11)

CFP

B = { aj,B} j = O,1, ••• ,n ( 1 2)

we obtain

CFP (A-B) = {aj , (A-B)} = {aj,A - a j , B} j = 0,1, ••

... ,n.

(13)

This state of affairs can now be regarded as follows:

The relative profitability of one of two mutually exclusive investment alternatives A and B can only be appropriately determined if the absolute profitability of the difference alternative (A-B) is examined.

If the difference alternative (A-B) is absolutely profitable, then alternative A is preferable to B • Otherwise, alterna-tive B is more profitable than A, which would mean the absolute profitability of the difference alternative (B-A) . These are completely general statements which fully apply to any decision criterion.

We should realise that, at the level of investigation of the relative profitability of two alternatives A and B , we are in fact investigating the absolute profitability of alter-native (A-B) , which means that at both stages of judgment, not only the absolute, but also the relative profitability, we shall make use of the same decision rules. Everything depends on them and we shall now look into them more closely with regard to both the decision criteria under discussion.

Net Present Value Criterion

According to the principle of this criterion, the absolute profitability of the difference alternative (A-B) means the fulfilment of the inequality:

n

L

a_{j ,} (A-B) . (1 + m) - j

~

0

j =0

If we take (13) into consideration, however, we obtain from (14): n

2::

(a. A - a . B)' (1 + m)-j

~

0 j=O J, J, and in continuation n _j L a . A' (1 + m) j=O J,

It is found that, in the case of the NPV ~riterion, the condition (14) for absolute profitability of the difference alternatives, can be replaced by direct comparison of the NPV's of both the competing alternatives by using relation

(16), which corresponds exactly to the customary procedure ( 1 4 )

( 15)

( 16)

when applying said criterion. This possibility became available only because the NPV of the difference alternative (A-B) is identical with the difference of the NPV's of the investment

alternatives A and B.

Internal Rate of Return Criterion

In the case of this criterion there is no such alternative possibility of application. As can be readily seen from the example of the two investment alternatives given in table 1, the Internal Rate of Return of the difference alternative

(A-B) will in general be distinguishable from the difference of the Internal Rates of Return of the two alternatives A and B that have to be compared.

Indeed, in the example referred to we have r_B

=

0.20 while, on the other hand, for r(A-B)

=

0.0954 ~ _{r A - r B·}

r

A

=

0.15 and CFP(A_B) we have

This makes i t clear that, in the case of the IRR criterion i t is not permissible to make a direct comparison of the IRR's of the two competing investment alternatives, which is none the less still done in the majority of cases.

We have just established in the foregoing that, in example 1, the two decision criteria NPV and IRR, provided they have been applied in the traditional manner, yield contradictory messages for m [0; 0.0954J If, on the other hand, we take the function NPV(A_B) of the difference alternative and consider its root (Fig. 2b), we recognise at once that in this case the message of the two criteria are in full agree-ment over the whole range

[0,

+ 00 ) •

This truth is by no means a recent discovery [4J,[10] ,[21], but i t has, alas, not been able to achieve general acceptance. Indeed, to date, apart from a number of quite sporadic cases

[15J, the opinion is still maintained that as well the agree-ment between the messages to be read NPV and IRR, as the application of the last-named criterion are only possible if the CFP's of both alternatives A and B, and the difference alternative (A-B) either have only one change of sign or, in the event that there are several such changes, have only one positive IRR. That this serious restriction of the area of application of the criterion is without object, is due to the incomplete, and in effect, incorrect interpretation of the criterion, as will be shown below.

4.2 £o~ple!e_d~f!n!t!o~ ~f_tge_I~R_c!i!e!i~

In the interpretation that has hitherto applied, the definition of the Internal Rate of Return, insofar as there exists such a rate, means finding the rate of return at which the NPV fun-ction is zero. Hence, one pOint on the interest axis will be sought whose value r is compared with the value m, the MAR assumed in the analysis. In dOing so, one is generally restric-ted to the cases in which the NPV function shows a classical

curve (Figs. 1a and 1b), these cases being based on a CFP with only one change of sign (expenditure followed by receipts) . The condition for absolute profitability of an investment alternative then reads as:

( 17) In the event that the CFP (with one change of sign) under

consideration showed the opposite sequence (receipts followed by expenditure), which can without a doubt occur in practice, the profitability condition would of course then have to be:

r ~ m • (18 )

If an NPV function has no real roots, or no positive root, or has more than one positive root, we are at a loss as to how to continue if we happen to be dealing with the IRR cri-terion in the customary way, since on the interest axis there is either no point or several pOints of comparison for our minimum profitability standard m.

In the first case there is talk of the phenomenon of non-exis-tence, in the second of ambiguity of the IRR criterion.

In both cases the critic cannot think of anything better than to issue an urgent warning against the use of the criterion and recommend the use of another instead (usually NPV) • The presumed failure of the IRR criterion in both these

"atypic" cases, while the NPV criterion is considered valid, should make one think twice, the more so as both

criteria are based on the same NPV function. This impression is strengthened by the fact that we have already succeded in filling up one ditch separating the two.

The real source of the misunderstanding is to be found this time in the not always permissibly simplified interpretation of the IRR criterion, which ultimately leads to the comparison of two pOints r a n d m on the interest axis. A complete definition of the profitability condition for this criterion will, on the other hand go as follows:

An investment alternative is only absolutely profitable in the sense of the IRR criterion when the MAR is to be found in a range [i 1 , i 2

J,

0

~

i1

<

i2 ' in which the NPV fun-ction of this alternative is non-negative, or when

NPV(m)

=

0 and NPV (m + e)

<

0

*

for an arbitrarily small

€

>

0 ) .

( 1 9)

If, on the other hand the minimum profitability m lies in the range [i3 ' i 4

J,

0

~

i3 <: i4 ' in which i t applies that NPV (i)

<::

0 , then the investment alternative being estimated is absolutely unprofitable.

As we can see, in this definition i t is the concept of a range instead of a pOint that comes to the fore, and this is a very significant difference from the interpre-tation of the IRR criterion that has hitherto obtained. This can be demonstrated by means of a very simple example, taking CFP(A_B) in Table 1 for the purpose. The NPV function of this CFP, to which r

=

0.0954 applies, can be seen in Fig. 2a . The profitability condition applying in this case

is given by inequality (17) and is fulfilled, for instance for m = 0.08 .

Instead of CFP(A_B) let us now consider CFP _{lB- A), which} differs from the preceding one only in that lts cash flows have been obtained by multiplying those of CFP(A_B) by -1. The function NPV(B_A) (i) is,of course, the m~rror reflection of function NPV(A_B) (i), and both functions have identical

roots r(A-B)

=

_{r lB- A)·}

If only two points r(A-B) and m had been exclusively relevant in making the estmate in accordance with the IRR

,,"

-criterion, we might have applied the profitability condition (17) in this case as well. Of course, this would be wrong, since the correct condition for this case is (18) as is evident from the circumstance that NPV(B_A) (i) ~ 0 for

i E [r(B_A) ,+00 J, just as (17) is the consequence of

*

_{) This last case can be considered as a border-line one in}

which the range [i1 ' i2J contracts to a single point i1

=

i2

=

m .

NPV (A-B) (i)

~

0 for i E:. [0, r (A-B)] •

As can be seen from this, the complete definition formulated above of the profitability conditions for IRR, though impli-cit in (17) and (18), has been pushed into the background in the course of their mathematical formulation,and forgotten. Now that the IRR criterion has been defined in full, let us explain the two "phenomena" referred to in the foregoing.

4.2.1 ~o~-~x!s~e~c~ 2f_n2n~e~a~i~e_I~R~

In this instance we are dealing with two fundamental cases. In the first i t is,CFP whose NPV function is always posi-tive throughout the whole range [0,+00) and in the second case, however, CFP whose NPV is always negative for

iE[O,+oO).

Now that we have linked the interpretation of the IRR cri-terion with a range instead of a pOint, the message from both cases is obvious and is, naturally, identical with that of the NPV. Thus, in the first place we have i1

=

0

and i2

=

+ 00, which means that the estimated investment alternative for optional mE:.[o,+oO) is profitable in the absolute sense. In the second case, on the other hand, we

have i3 = 0 and i4 = + otO , which means that the alternative under consideration for optional m ~

[0,

+ 00) is not profi-table in the absolute sense. In both the cases mentioned, provided they are recognised early enough, the investor need not trouble about the height at Which the MAR should be set.

A trivial example can be used to illustrate the two cases in pOint, that is a CFP in which cash flows with the same sign occur exclusively.

Table 2 gives a different example, to wit, from the CFP's of two investment alternatives A and B , which have IRR's with r

A = 0.30755 and rB = 0.3000, the difference alter-native is obtained, whose CFP - as can be proved - generates a constant positive NPV function in spite of four changes

of sign over the whole range [0, + 00) (Fig. 3) . TABLE 2 j 0 1 2 3 4 CFP A -9000 +1200 +10900 +1200 +3420 CFP B -10000 +6000 +1100 +10000 +520 CFP (A-B) +1000 -4800 +9800 -8800 +2900 4.2.2 ~x!s!e£c~ ~f_s~v~r~l_p~s!t!v~ !R!~~

It has long been known that CFP~s having more than one change of sign are potentially capable of creating NPV functions with more than one real root, so that CFP~s can have more

than one positive IRR.

In the light of our definition of the IRR criterion, the in-terpretation of such a case, faced with great fear and trepi-dation by all analysts up to now, is really very simple. Nevertheless, before proceeding to the task, let us first pass in review the phases of the so-called ambiguity of the IRR criterion, using a simple example for the purpose of arriving at a better understanding of the matter.

We commence our example with a classical CFP comprising three expenditures and the same number of receipts, presen-ted in Table 3 as CFP

1. This CFP generates an NPV function which, as we already know (Fig. 1b) always has one root and one minimum.

Further, starting from year 6, we intend to add more cash flows to this CFP succes~ively, all of which are negative. In this way six new CFP~s come into being: CFP

2 - CFP7 ' all of which are contained in lines 2-7 of Table 3.

What the newly generated CFP~s have in common, apart, from the fact that they all have the first six cash flows identical, is that two changes of sign occur in consequence of their cash flows. This means that all these CFP~s can also have two IRR~s and they do have them in fact, as is shown in

Fig. 4 , which gives the curves of the NPV functions of all the CFP's discussed. The characteristics of these functions are given in Table 3, in which r(k) and r(k) are the

01 02

roots of the kth NPV function. Columns 17, 20 and 21, 22 con-tain the coordinates of the appropriate minima and maxima. As we assume in our example a MAR of m

=

0.10 I we also

computed the corresponding NPV's shown in column 16. In the light of this description let us now consider the

diagrams of the NPV functions in Fig. 4 somewhat more closely. We note that, after having added the cash flow -200 in the

6th year to CFP

1 ' so that CFP, became CFP2 ' function NPV₁ (i) in its transition to function NPV

2(i) underwent a ra-dical change in the left part of its curve, the cause being the different limiting values of both functions for i~-'

(compare relation (3». As a result, function NPV

2(i) , com-pared to NPV, (i), acquired an additional root within the

range of the negative value of i ; function NPV₂(i) acquired a maximum at the same time.

The other NPV functions, that is those of CFP

3 - CFP7 assume similar forms to those of functions NPV

2(i).

As is shown in Table 3, the successive addition of eVer-new cash flows has the result that as well the roots of the

corresponding NPV functions as the abscissae of the relevant maxima and minima approach one another more and more closely. After this description let us turn our special attention to

the roots (IRR's) of the seven NPV functions. We see that so long as the algebraic sum of all cash flows of a CFP is

positive, the NPV function of this CFP has only one positive root or a positive and a negative root. In our example this applies to the first four CFP's. If we were, for a moment, to assume the standpoint obtaining in the traditional inter-pretation of the IRR criterion, we could be quite satisfied with these first four CFP's. Each of them has only one

alterna-k CFP k a k ,J . j = 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9 10 1 CFP 1 -400 -500 -600 +1800 +1400 +2200

-

-2 CFP 2 -400 -500 -600 +1800 +1400 +2200 -200

-3 CFP 3 -400 -500 -600 +1800 +1400 +2200 -200 -1200 4 CFP 4 -400 -500 -600 +1800 +1400 +2200 -200 -1200 5 CFP 5 -400 -500 -600 +1800 +1400 +2200 -200 -1200 6 CFP₆ -400 -500 -600 +1800 +1400 +2200 -200 -1200 7 CFP₇ -400 -500 -600 +1800 +1400 +2200 -200 -1200 a k . (k) _r(k)

i~~~

,J _NPV k (0.1) k r 01 02 8 9 10 1 1 12 13 14 15 16 17 18

-

-

-

-

0.5497 +2324.20

-

1

-

-

-

-0.9143 0.5417 +2211.30 -0.8969 2

-

-

-

-0.3848 0.5066 +1595.51 -0.2691 3 -2400

-

-

-0.0108 0.4410 +475.90 +0.1177 4 -2400 -100

-

0.0000 0.4384 +433.49 +0.1271 5 -2400 -100 -500 +0.0482 0.4289 +240.72 +0.1673 6 -2400 -100 -1500 +0.1290 0.4054 -144.83 +0.2283 7 k NPV (i(k» _~ . (k)

_.

NPV ( i (k) ) k max mJ.n k min 19 20 21 22 1

-

2.0691 -540.49 2 36302445.77 2.0616 -540.73 3 5774.64 2.0441 -541.22 4 481. 14 2.0304 -541.55 5 445.76 2.0301 -541.55 6 319.22 2.0297 -541.56 7 154.95 2.0289 -541.58

tives being judged are all of them absolute profitable. Difficulties crop up, however, at CFP

5 and cause complete chaos in the next two. The reason for this is that, in all three last CFP's, two non-negative roots suddenly make their appearance, with the result that we come up against the

ambiguity problem of the IRR criterion. If we were to follow the advice given in orthodox quarters on such cases, we

should be obliged to dispense reluctantly with the controver-sial IRR criterion in the three awkward cases.

We would, however, be entitled to ask how i t is possible that the mere adding of a single cash flow of Fl. -100 *) to

the CFP4 should have such far-reaching consequences

which suddenly question the whole philosophy of an important decision criterion. Inevitably the suspicion is aroused that there must be something amiss with it.

Such problems do not arise i f we take the interpretation of the IRR criterion put forward here as the pOint of departure.

For k=1,2, ••• ,6 the MAR m=0.10 lies within ranges in which the corresponding NPV functions are positive. In these cases we have i

1=0 for k=1,2, ... ,4 and

i1=r6~)

for k=5,6 and

i2=r6~)

for k=1,2, ••. ,6 , which means that all of the first

six investment alternatives in the sense of the IRR criterion are absolutely profitable.

It is only in the case of k=7 , which has come into being by replacing the last cash flow -500 of CFP6 by the cash flow -1500 , that the MAR m leaves the zone of positive values of function NPV₇(i). As a result we have in this case i3=0

d . (7)

an ~4=r01'

From Fig.4, based on the numbers in Table 3, i t is to be seen that both investment criteria, NPV as well as IRR, give iden-tical information. This will always be the case because the identicalness arises simply out of the complete definition of the IRR criterion.

*) It is, of course, perfectly easy to construct an example

in which the same effect can be brought about by an arbitra-rily small sum of money.

Let us now once again follow all seven NPV functions in their order of generation and observe the way in which the profita-bility of the investigated alternatives is successively reduced. In the NPV this becomes apparent in the falling value of the criterion, whereas in the IRR criterion the loss of profitabi-lity arises out of the constant narrowing of the range [i 1 ,i2

J,

the left end of which is extremely mobile.

It should be noted in the context of the example discussed above, that both types of CFP occurring in it are classical in evaluating investment projects and are therefore very realis-tic.

The first type of CFP, which starts with expenditure and has only one change of sign, has already been discussed in the present study when it was established that these two phases of the CFP correspond to two fundamentally different periods encountered in the operational life of an investment project - the initial period of investment and the subsequent one in which returns are obtained.In the second type of CFP, which we shall also regard as classical, a third phase can be observed, that of negative cash flows and with it an additional, that is to say second change of sign. This phase is always to be found where, in the course of a project, expenditures arising from various causes have to be reckoned with at the end of the industrial or service activities that were envisaged in the project. Thus, for example, at the end of the extraction phase in opencast mining operations, expenditures required for

recultivation of the devastated area concerned, or expenditures incurred in the liquidation phase of an enterprise, against which there are no or very few receipts to be expected.

In general, it can be taken as a starting pOint that, in the course of time and of the growing protest against devastation of the environment, CFP's of the second type will have to be made use of more and more often.

The example put forward made it clear that if the IRR criterion is interpreted according to its actual sense, a so-called case of ambiguity of this criterion will be quite a normal one, assessment of which will not give rise to difficulties of in-terpretation. Two further examples will be adduced to show that

there are no such difficulties in evaluating the relative profitability of two alternative investments whose NPV fun-ctions have several positive IRR's.

Let us consider an example of CFP's from two investment alter-natives A and B shown in Table 4.

TABLE 4 a. A _{J ,} CFP ,B , (A-B) j

=

₀ 1 2 3 CFP A -10000 +43000 -60400 +27720 CFP B -2000 +9300 -14050 +6900 CFP(A_B) -8000 +33700 -46350 +20820

The CFP's A and B of the two alternatives A and B , as well as the difference alternatives (A - B) arising therefrom, generate NPV functions whose curves are shown schematically

in Fig. 5, and their characteristics given in Table 5.

As can be seen, the NPV functions of all three alternatives each have three positive roots, which means that the semiaxis of the interest [0,+ oa) is subdivided into four zones with constant signs of the appropriate NPV function. The decision as to whether an alternative is absolutely profitable or not, now depends directly on which zone the MAR m is to be found in. For instance, alternative A is profitable within the ranges [0,0.10J and [0.40,0.80J I and is not profitable in the ranges

(0 . 1 0 , 0 • 40 ) and ( 0 • 80 , + 00 ) .

In alternative B, on the other hand, the profitability zones are the ranges [0,0.15J and [0.50,1.0J, and this alternative is absolutely unprofitable in the ranges (0.15,0.50) and

(1.0,+oC).

native ,B _,(A-B) ,B _{, (A-B) , (A-B)} ,B ,B _,(A-B) B ,B (A-B) , (A-B) A 0.10 0.40 0.80 0.2348 -66.857 B 0.15 0.50 1.00 0.2729 -19.697 (A-B) 0.075 0.3687 0.7688 0.1798 -56.832 TABLE 5 (contin.) i _max,A NPVA(i A) NPV_A (0.45) Alter- max, native ,B _,(A-B) B ,B B

(A-B) ,(A-B) (A-B)

A 0.5745 +47.838 +20.091

B 0.7486 +13.995 -5.412

by the absolute profitability of the difference alternative, we have in our example for the relative profitability of alter-native A the following MAR zones: [0,0.075) and

(0.3687,0.7688), and for the relative profitability of alter-native B, on the other hand, the ranges (0.075,0.3687) and

(0.7,+oC). For the MAR"s 0.075,0.3687 and 0.7688 the two competing alternatives are equally attractive.

However, we have not yet finished with the problem of estimating the relative profitability of alternatives A and B as we have still to ensure that the relatively profitable alternative is also absolutely so. Viewed in that light, we still have to check the zones of relative profitability of both alternatives as to the extent to which they meet the demands of absolute profitability. The result is as follows:

1. The investment alternative A is absolutely and relatively profitable if the MAR lies within one of the two ranges:

[0,0.075) or [0.40,0.7688).

2. The investment alternative B is absolutely and relatively profitable if the MAR lies within one of the two ranges:

(0.075,0.15] or (0.7688,1.0] •

3. The two alternatives are equally attractive, both in the absolute and the relative sense, if the MAR takes the values 0.075 or 0.7688.

4. Neither of the two alternatives is simultaneously profi-table in the absolute and relative senses if the MAR

lies in either of the ranges: (0.15,0.40) or (1.0,+ c>O).

This situation has been represented graphically in the lower part of Fig. 5, neglecting proportionality, however. If the MAR were to be 0.45, for instance, i t would be found in the range [0.40,0.7688), which would mean that according to the IRR criterion the investment alternative A is profitable both in the absolute and relative senses.

The same result is, of course, to be found for the NPV criterion in the last column of Table 5.

the NPV function of the difference alternative, but also those of the two investment alternatives had more than one positive root, that is the IRR criterion was ambiguous here at both decision stages in the traditional sense. It is perfectly possible, however, and will probably occur more often in prac-tice, that the CFP's of the two alternatives under comparison generate "conclusive" NPV functions as regards the IRR which then lead to "ambiguous" NPV functions of the difference alter-native. Such a case is illustrated in Table 6, in which the corresponding CFP's are given, and in Table 7, where the NPV functions of these CFP's are characterised, as well as in Fig. 6, in which the case is presented in graphic form. To close this chapter we should like to observe in addition that it is quite immaterial, when forming the difference al-ternative, which of the two competing alternatives we take as alternative A and which as B. If, for example, we were to ex-change the roles of the two alternatives A and B we would in the end come up with a difference alternative (B - A) whose NPV function NPV(B_A) (i)

= -

NPV(A_B) (i) gives identical

information on the ranking of the two alternatives, just as was given previously by the function NPV(A_B) (i).

TABLE 6 a. A _J, CFP ,B , (A-B) j

=

0 1 2 3 CFP_A -4000 +2500 +460 +14400 CFP_B -3000 -3100 +10550 +8550 CFP(A_B) -1000 +5600 -10090 +5850

Alter-native ,B ,B ,B ,B

,(A-B) , (A-B) , (A-B) ,(A-B)

A 0.80

-

-

-B 0.80

-

-

6.8592 (A-B) 0.30 0.80 1. 50 0.4664 TABLE 7 (contin.) imax,A NPVA (i A) Alter- max, native ,B B ,B

, (A-B) (A-B) ,(A-B)

A

-

-B

-

-(A-B) 1.1372 10.4918 B ,B (A-B) ,(A-B)

--3206.0262 -18.1959 tv U1

5. Net Present Value or Internal Rate of Return?

It has been demonstrated in the course of the present dis-cussion that, given the correct procedure during assessment of investment alternatives and correct interpretation of the

IRR criterion that,

1. both decision criteria, NPV and IRR, lead to identical results in all conceivable decision situations;

2. no applicational limits are set for the IRR criterion (for instance, the occurrence of "ambiguity" in its results) and, should they have to be set for any reason, they would be equally applicable to the NPV criterion, as the IRR criterion is based on the same philosophy as that of the Net Present Value.

The question thus arises as to whether both criteria are equivalent in rank as regards assessment of investment alter-natives, or whether one or other is the more suitable.

To answer this question we must first of all realise the extent to which the answers given by each of the criteria satisfy our requirements. For instance, when the assessed alternative is absolutely profitable we recognise that using NPV criterion by the non-negative value of the NPV function of said alternative for the MAR m (Fig. 7a and 7b).

The IRR criterion in this case, on the other hand, will not only give us the same answer, but some important information in addition. It tells us how wide the range is of the constant sign of the NPV function in which the MAR m is situated, and where both boundaries are located. From this we learn how large the "returns bolster" is in the upwards and downwards directions separating m from both ends of the range in which the NPV function with constant sign is to b~ found (Fig. 7a and 7b).

In view of the fact that the determination of MAR is obtained by means of a more or less rough estimate, the actual objective value of MAR must always be expected to deviate somewhat from that of m assumed in advance in the c~lculation.

In that situation the IRR criterion enables us to turn our thoughts to the stability of the decision taken. This cannot be done with the NPV criterion, however, as there we only have one single value of the function available and know nothing further about its behaviour in the vicinity of the MAR (Fig. 7a and 7b).

This situation leads us to assume that, fundamentally consi-dered, the IRR deserves preference over the NPV criterion. On the other hand, it must be admitted that Net Present Value

is a decidedly more "convenient" and "elegant" criterion as, has already been said, consideration of the difference value can be dispensed with when the NPV criterion is applied. Determination of the Net Present Value is also easier than searching for the roots of the NPV function, although this argument is steadily losing ground in view of such efficient computational aids as computers or specially designed mini-computers.

The choice between the two criteria is all the more difficult, as they complement one another to a large extent, as can be seen from Figs. 7a and 7b.

In Fig. 7a we have the sections of two NPV functions, NPV₁ (i) and NPV2 (i) which indeed have an identical Net Present Value for i=m, but where the function NPV₁ (i) has a much wider constant sign range and thus gives a more stable result than is the case for function NPV

2(i). This can only be recognised, however, when both the IRR and the NPV criteria are applied. The opposite situation is found in Fig. 7b, where b9th NPV functions have two identical roots between which these fun-ctions have the same sign. The IRR criterion would, in both decision situations, fail to allow any difference to be reco-gnised, and would require the inclusion of the NPV criterion to enable a differentiation between the two criteria to be made.

The peculiarity of the two criteria discussed above that, in fact they lead to identical decisions in identical decision situations, but that there is a difference in the information

they convey, can lead to questionable decisions if one were to depend exclusively on one of the two criteria.

We shall make this clear by an example in which the MAR turns out simultaneously to be the Internal Rate of Return of a CFP. Such a case, which can be regarded as a border-line case in which the alternative to be assessed can be taken as just acceptable, can occur in four different ways, as is to be seen in Figs. 7c to 7f. If we were to be led exclusively by the NPV criterion, we should be unable to distinguish between the four eventualities. Among other things this would mean that, in two curves of the NPV function as shown in Figs. 7e and 7f, we should be inclined to react as identically positive, which would be justified only in the 7e case.

If, on the other hand, we were also to make use of the IRR criterion, which assumes investigation of the roots and beha-viour of the NPV function in the root range, we should inevi-tably recognise that a positive decision in the case 7f would be extremely risky becouse of its instability.

The foregoing arguments have shown clearly that it would be unwise to depend on only one of the criteria under discussion. The dilemma used as the heading of this chapter was intended rather to be solved by using both criteria together in the economic assessment of investment alternatives. This rule must apply particularly when there are indications that doubt-full cases are likely to occur.

The acceptance of the solution derived means inevitably that the main role in seeking the decision falls to the investi-gation of the NPV functions of both the investment and dif-ference alternatives. It is true that this involves more cal-culation compared with customary practice to date. However, deeper insight into the decision situations is obtained there-by and the decisions which follow can be given a more solid foundation.

6. Conclusions

From the content of the present study the following conclu-sions can be drawn:

1. The basis for assessing the relative profitability of two mutually exclusive investment alternatives is repre-sented by the CFP of the difference alternative.

The information on the absolute profitability of the difference alternative furnishes at the same time the judgement as to the relative profitability of the two alternatives submitted for comparison.

This principle applies in particular to the IRR criterion while, in the case of the NPV criterion, direct

compari-son between the competing alternatives also furnishes correct result.

2. One of the causes of misunderstanding in the actual use of the IRR criterion arose from imperfect interpretation, which led to false assessment of the criterion and degra-dation of its importance in investment calculations. It was shown that IRR, properly interpreted, furnishes identical results and, as regards informational content, merits a higher ranking than the NPV criterion.

3. By filling in the gaps in the interpretation of the IRR criterion, the cases of non-existence or ambiguity of the Internal Rate of Return and hence the assumably limited application of the criterion were explained away. The basis for the far-reaching objections to this criterion was therefore destroyed.

4. As the difference alternative is intended to act as the

basis for application of the IRR criterion, "the weightings" of the IRR's of both the investment alternatives will

also be taken into account. This will also set aside yet another objection to the IRR criterion.

5. In the interpretation put forward in these pages, the IRR criterion can serve as a decision criterion in all those cases where the NPV criterion has been regarded as exclusively authoritative, for instance optimisation of all investment programmes, or determination of the optimum replacement moment, to name but two examples. 6. Both decision criteria, NPV and IRR, furnish information

which leads to identical decisions, but whose informa-tional contents are complementary. For that reason it is advisable and, in order to be able to recognise pro-blematic decision situations, indeed necessary to make use of both criteria at the same time.

This, in the end, boils down to examining the correspon-ding NPV functions during the preparatory stage of an investment decision, as has been done in the examples discussed here. This plainly makes the NPV function the most important decision instrument in judging investment alternatives.

7. In view of the need to examine the NPV functions in the course of the decision process, the same objections can arise in the minds of the scientists concerned as have in the past so often and so urgently been put forward with reference to the IRR criterion, namely that the procedures for determination of the roots of the NPV function and the examination of the function were very time-consuming.

That may have been the case in the past, but now, however, in view of the multiplicity and efficiency of computatio-nal aids and methods available, such arguments are prac-tically without foundation. In very many applications standard computer programs can be used or a special one written. Those with no acces to a computer can, without much trouble, in fact sometimes even more efficiently, make use of a specially oriented programmed mini-reckoner

(such as the HP-38E) or the programmable TI-58C, for instance.

BIB L lOG RAP H Y

[1J Bacon, P. W., "The Evaluation of Mutually Exclusive Investments", Financial Management, 6(1977), Summer pp.55-58

[ 2] Bernhard, R. H., "Discount Methods for Expenditure Evaluation - A Classification of their Assumptions", The Journal of Industrial Engineering, 13(1962), January-February, pp.19-27

[3J Biergans, E., "Kritische Bemerkungen zur Kritik am internen ZinsfuB", Betriebswirtschaftliche Forschung und Praxis, 25(1973), Mai pp.241-261.

[4] Blohm, H., K. Liider, "Investition", 4.Aufl., Verlag F. Vahlen, Miinchen 1978

[5J Brigham, E.F., "Hurdle Rates for Screening Capital Expenditure Proposals", Financial Management, 4(1975), Autumn, pp.17-25

[6J Chambers, J.C., S.K. Mullick, "Investment Decision-making in a Multinational Enterprise", Management Accounting, 52(1971), August, pp.13-20

[7] Dam van, C., "Capital Investment Decisions: An Intro-duction", in: Dam van C. (edit.), "Trends in Financial Decision-making", Nijenrode Studies in Business", VOl.2, Leiden/Boston 1978, pp.203-208

~]

Diepenhorst, A.I., "Some Features of Capital Budgeting within Three Leeding Dutch Companies", in: Dam van, C.

(edit.), "Trends in Finantial Decision-making", Nijenrode Studies in Business",Vol.2, Leiden/Boston 1978, pp.249-260

[~

Enden van der, C., "Beslissingscalculaties", Samson Uitgeverij, Alphen aan den Rijn-Brussel 1975

[10] Fleischer, G.A., "Two Major Issues Associated with the Rate of Return Method for Capital Allocation: The

"Ranking Error" and "Preliminary Selection", The Journal of Industrial Engineering, 17(1966), No.4, pp.202-208

[11J Fremgen, J.M. , "Capital Budgeting Practices: A Survey", Management Accountin2, 54(1973), May, pp.19-25

[12] Gitman, L.J., J .R. Forrester jr., "A Survey of Capital Budgeting Techniques Used by Major US Firms", Issues in Mana2erial Finance, Hinsdale, Illinois 1980

[13] Hajdasinski, M., "Jednoznacznosc kryterium wewnetrznej stopy procentowej w przypadku najogolniejszej postaci strumienia wartosci 0 jednej zmianie znaku" (Uniqueness of the IRR Criterion in Case of the Most General Form of the Cash-Flow-Pattern with Single Change of Signe),

Zeszyty Naukowe AGH, Nr.380 Krak6w 1981

[14J Klammer, T., "Empirical Evidence of the Adoption of Sophisticated Capital Budgeting Techniques", Journal of Business, 45(1972) July, pp.387-397

[15J Leautaud, J .L.L., R. Swalm, "On the Fundamentals of the Economic Evaluation", Engineerin2 Economist, 19(1974), Winter, No.2, pp.105-125

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D~

Merrett, A.S., A. Sykes , "The Finance and Analysis of Capital Projects", J.Wiley 1963

[1~

Pares, A., "Survey on the Application of Analytical Techniques in Investment Decisions: France 1974", The European Institute of Business Administration (INSEAD), Research Paper Nr.E156, Fontainebleau, 1974

[1~

petty, J.W. O.D. Bowlin, "The Financial Manager and Quantitative Decision Models", Financial Management, 5(1976), Winter, pp.32-41

[2

oJ

Petty,J.W., D.F. Scott jr., M.M. Bird, "The Capital Expenditure Decision Making Process of Large Corpo-rations", The Engineering Economist, 20(1975), No.3, pp.159-172

[ 21J Thuesen, H. G., W. J. Fabrycky, G. J. Thuesen, "Engineering Economy, 5th Edition, Prentice Hall, Inc. 1977

"

L

NPV

k

(i)

-1

-1

I

I

I

,

I I 0

:0,08

I I

0.0954

I I

-1000

-2000

N'PV

o

I , .

.,»,.

0,095it

-180

~Z" 16

/ 1/

-1000

I I

-.

L (') ~ll'Jt~-A) l _ _ _ _ _ _ _ _

....

---.",-

...

---.

L

NPV(A_8)(t)

+

1000

-f

NllV ..

(i)

NPV.z.

(i)

NPV

_s

(i)

NPV,

(D

NPV;t(l)

• l

J \

La- \

j-O Jol

a

A profitable both abso-lute and relative B profitable both abso-lute and relatl.ve A and B unprofita-ble either absolute or relative 0

4,

(A-J) ~, (A-a)

-

....

_...

...

"

_{" " r;,»}

•

i.

•

L

r

1,A r;,<A-&) ~,a r~, (A-I) r~,B

J:.,I

r.a,A ~.& +00

----1

\

,

\ \ 3

LQJ'~

j=O

J

o

\

,

NPVBCO

,

"

...

~"

L Qo.a

Q--t---+-.;:!Io",~=-.._-_-_-

---

aO.A~~---r---~~==~~~~---"

F-r

aj.(A-B)

r

_{UA -.)} L

.

QO.(A_B)o--t---:::!""""=::::::::..o"c....---+---A profitable

both abso-

0

~)(A-B)

lute and t-~---+---relative B profitable both abso-lute and relative A and B unprofita-ble either absolute or relative

'4.

(A-8) ~.A ---+0<:

Nl'Vz(i)

_{NPV-1 (0}

I / I

"

I \ I

~

\ I \ I

NPV,a

(m) I \

--

...

-~

...

I \ - I -ra,~ I

.

\ rz.

_{• J} 2 L •

0

.",~.--

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... I _m " ,

m

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NPV

NPV(l)

• • I.

o

t.

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.

.

L

o

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L

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RESEARCH REPORTS

REPORT NR

ISBN

EUT/BDK/l

90-6757-001-X

MirosJaw M. Hajdasinski, "Internal rate of

return - an investment decision criterion of

full app

1 i

cabil

i ty"

EUT /BDK/2

90-6757-002-8

L.A. Soenen, IIA portfol i

0

approach to the

capital budgeting decision"

EUT/BDK/3

90-6757-003-6

R.J. Kusters, IIPatient scheduling: a review"

EUT/BDK/4

90-6757-004-4

P.W. Huizenga, C. Botter, I1Researchinspanningen,

technische innovatie en

werkgelegen~eid:

Een

internationaal vergelijkende studie door middel

van research-indicatoren"