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THE WEIGHTED INTERNALRATE OF RETURN (WIRR)AND

THE EXPANDED BENEFIT- COST RATIO (EB/CR)

Ignacio Velez-Pareja

ivelez@javeriana.edu.coDepartment of Management

Universidad JaverianaBogot, Colombia

Working Paper N 9First version: September 19, 2000This version: October 22, 2000

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THE WEIGHTED INTERNALRATE OF RETURN (WIRR) AND EXPANDED BENEFIT-COST RATIO (EB/CR)

Ignacio Velez-Pareja

ivelez@javeriana.edu.co

ABSTRACT

The purpose of this paper is twofold: the first purpose is to tell a nave history of

an experience in constructing knowledge. In this first part I tell how I approached the

process to solve the inconsistency between the net present value, NPV, the internal

rate of return, IRR and the benefit-cost ratio B/CR or profitability index. There are

improvements and drawbacks, mistakes and successes. I think this is important for

our students because sometimes they might believe that discovering new ideas or

new approaches is an insurmountable job. Sometimes it is, but they have to learn

that this is a cumulative process with advances and drawbacks and sometimes it is

necessary to start again and start from scratch. The second purpose is to study the

development of a procedure to include the implicit assumptions ofNPVin the IRR and

the profitability index (benefitcost ratio B/CR). The resulting indicators are named

the weighted IRR (WIRR) and the expanded B/CR (EB/CR). These two desirability

measures have the property to coincide with theNPVranking for investment analysis

and hence, will maximize value. Examples are presented.

KEYWORDS

Net present value, NPV, internal rate of return, IRR, benefitcost ratio, B/CR,

profitability index, NPV assumptions, overall rate of return, modified internal rate of

return, MIRR.

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INTRODUCTION

The contradictions between NPV, IRR and B/CR1

are well known in the financialliterature (see Lorie and Savage, 1954, Grant and Ireson, 1960, Fleischer, 1966,

Bacon, 1977; Oakford, Baimjee and Jucker, 1977, Canada and White. 1980,

Beidleman, 1984, just to mention a few.) The usual procedure to solve this conflict is

to calculate the incremental IRR and the incremental B/CR(Grant and Ireson, 1960,

Bernhard, 1989). However, this procedure is not easier to understand than NPVand

the intuitive appeal ofIRR and B/CR is a strong incentive for practitioners to keep

using them. Others (Lin, 1976, Beaves, 1988 and Bernhard, 1989,) have proposed

modified IRR s taking into account the reinvestment of intermediate cash flows. Velez-Pareja (1979a, 1979b, 1979c and 1994) presented an approach that modifies theIRR

and the B/CR, (defined as the present value of all the inflows or positive cash flows

divided by the present value of all the outflows or negative cash flows) and takes into

account the implicit and explicit assumptions in NPV. Shull (1992) starts from the

modified IRR proposed by Beaves (1988), Bernhard, (1989) and Lin (1976) to

construct the adjusted overall rate of return. This paper studies the method

proposed by Shull and proposes the method to modify the B/CR in order to make

them coincide with the NPV ranking. This time a formal proof is presented for both ofthem. Emphasis is made in that it is necessary to make NPV and IRR compatible by

the explicit inclusion of the implicit assumptions ofNPV in the IRR. In this article the

Shulls adjusted overall rate of return will be illustrated. The expanded benefit-

cost ratio is presented and both methodologies are illustrated with examples to

show how they are consistent with shareholder wealth maximization.

SOME BACKGROUND AND HISTORY

When a student is working on an essay or a thesis I recommend that she registers

the advances and drawbacks found during the process of doing her research. The

1The benefit-cost ratio (also called profitability index) is defined as the ratio between the present value of

cash inflows and the present value of the cash outflows.

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same will be done in this section. I will tell a nave story of how I arrived at the

findings I am presenting in this short paper.

Thinking and studying about the contradictions between NPV, IRR and b/cr

rankings I concluded that their origin is the difference in assumptions of the

different methods. I concluded, then that there are two assumptions to be

considered:

1. Regarding the reinvestment of intermediate cash flows, and

2. Regarding the amount to be invested in each alternative

THE REINVESTMENT OF INTERMEDIATE CASH FLOWS

As was mentioned in another paper (Velez-Pareja, 1999a) the NPV is just a

model that tries to approximate as much as possible to reality.

A financial manager or a treasurer never leaves the cash surplus idle in a bank.

She will invest it to gain some return. In this sense, the NPV considers that

intermediate cash flows are reinvested and the mathematics involved in the

discount process implies that the reinvestment is done at the same rate the cash

flows are discounted. In the just mentioned paper it is shown that in reality what

is reinvested is the surpluses of cash, not the free cash flows. However, this canbe considered as a limitation of the model. When working with free cash flows

and pro-forma financial statements, including the cash budget, this reinvestment

is made explicitly at the expected opportunity rate of the market. On the other

hand, the IRR assumes implicitly that the reinvestment of intermediate cash flows

is made at a rate equal to the same IRR , while the B/CR assumes that the

reinvestment is done at the same rate of discount.

THE AMOUNT TO BE INVESTED IN EACH ALTERNATIVE

When analyzing mutually exclusive alternatives it is assumed that there exist

funds to invest in any of the alternatives considered. This means that the firm

has available the funds needed even for the largest alternative. Then, there is a

hidden assumption regarding the investment of the difference between the present

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value of the investment in the actual alternative and the alternative with the largest

present value of the amount invested. And this implicit assumption usually is not

mentioned in the literature. If this is so, every time a smaller alternative is

analyzed, there will be some funds in excess. These funds might be invested at

the opportunity cost found in the economy. (In fact, if the project is financed withexternal funds, it means that the debt will be smaller). This difference is invested

at the same rate of discount used to calculate the NPV. The NPV of this extra

investment is just zero. This means that each alternative could be seen as composed

of two cash flows: one the cash flow associated to the alternative itself and a second

one, the cash flow of the difference in the amount invested. This later cash flow is

composed of just two flows of funds: one outflow at the beginning (the difference in

amount invested, K-I, where K is the total funds available in present value and I is

the present value of the amount to be invested in the current alternative) and an

inflow at period n equal to (K-I)(1+i)n. The NPV considers implicitly that those funds

are invested at the rate of discount and its NPV is equal to 0. On the other hand,

the IRR and the B/CR say nothing about this difference. In fact, as relative

measures, IRR and B/CR are blind to the amount invested. (Just recall that you can

get an IRR of 500% either if you invest $1 today and receive $6 in a year or if you

invest $1 million and receive $6 million in a year).

The NPV is an absolute, not a relative measure of the value added by the project to

the firm. This means that it takes into account the amount invested as part of the

value calculated. In fact, the NPV subtract the amount invested from the net benefits

discounted at time 0. On the other hand, IRR tells if there is some value added, but it

never says how much value has been created.

The usual assumption mentioned in most finance textbooks is the reinvestment of

intermediate cash flows at the discount rate, when NPV calculations are made.

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SOME HISTORY

In 1979 the author wrote some class notes for the students of a course in

Investment Decision Analysis.2 In those notes an attempt to introduce the implicit

assumptions ofNPV into IRR and B/CR was made. This first approach considered tofind an arithmetic weighted IRR , taking into account the modified IRR (with

reinvestment at the discount rate) weighted by the amount invested in the

alternative (I) and the discount rate weighted by the extra amount (K-I). It was an

arithmetic average. In a working paper (a recompilation of class notes) for faculty

and student use3, a comment was published indicating that a geometric average

might be calculated adding up the cash flows. However, tested with an example it

was found that the results were basically the same and the ranking did not change.

For this reason, the approach of arithmetic average was left as it was. In a posterior

recompilation4, the abovementioned original class notes were published.

In 1989 a paper with these ideas was submitted to The Engineering Economistand

it was rejected. A proof was not presented in order to check if the ranking obtained

by the weighted average IRR and the expanded B/CR coincided with the ranking

provided by the NPV. However, Monte Carlo simulations were made and the results

were satisfactory (Echeverri, 1987, Ochoa, 1987).

The original proposal for the weighted internal rate of return (WIRR) was

K

IKiIIRRIRR R

w

)( += (1)

where IRRW is the arithmetic weighted internal rate of return, IRRRis the internal rate

of return with reinvestment of intermediate cash flows at the discount rate,Kis the

total funds available in present value, Iis the present value of the amount to be

invested in the current alternative and iis the discount rate.

2Velez-Pareja, Ignacio, Toma de Decisiones, Alternativas de Decisin y Algunos Mtodos para su

Ordenamiento y Seleccin, Universidad de los Andes, Industral Engineering Department, 1979, mimeo, 48

pp.3

Velez-Pareja, Ignacio, Decisiones de Inversin: Textos Preliminares, Universidad de los Andes, Industrial

Engineering Department, August, 1979, mimeo, 116 pp.4

Velez-Pareja, Ignacio, Decisiones de Inversin: Textos Complementarios, Universidad de los Andes,

Industrial Engineering Department, December, 1979, mimeo, 209 pp.

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This approach was kept up to the publication of Velez-Pareja, 19945.

After several years working with the private sector, I returned back to the

academic life. In the first term of 1995 it was found a case where the consistency

between the ranking with NPV and the ranking with the weighted IRR did not work.

During the same term, as a result of an assignment to my students, an article by

David Shull published in The Engineering Economistwas found. In that article,

published in 19926, Shull presented a yield based methodology "that provides capital

project evaluations consistent with shareholder wealth maximization". This

methodology considered the weighted geometric average of the IRR with reinvestment

(the modified internal rate of return, MIRR) and the discount rate. Shull defined this

rate as the adjusted overall rate of return.He offered a proof of consistency between

the behavior of this indicator, the NPV acceptance/reject decision and the NPV

ranking of investment alternatives.

This method might present mathematical problems for extreme cases, but it is

good enough to solve common problems of inconsistencies between NPV and IRR .

THE IMPLICIT ASSUMPTIONS IN NPV CALCULATIONS

The idea in this paper is to include explicitly both assumptions in the IRR andB/CR calculations.

A word has to be said on why to include the assumptions of the NPV in the IRR

and the B/CR and not the contrary. First, the NPV measures the value created by

an alternative for the firm (see, Velez-Pareja, 1999b). The IRR is a relative measure

of the amount of resources generated by the amount invested (this is not a

measure of the value generated by the alternative) and the B/CR measures how

many times the alternative generated the amount invested (it is not either a

measure of the value generated). Second, the assumption to reinvest the cash

flows at the rate of discount is more plausible than to reinvest the cash flows at

the IRR. The rate of discount is defined from actual interest rates available in the

5From February 1988 to May 1994 I was not working at an university.

6Shull, David M. Efficient Capital Project Selection Through a Yield-Based Capital Budgeting Technique,

The Engineering Economist, Vol. 38 No. 1, Fall 1992

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market (the cost of debt and the cost of equity, see Velez-Pareja 1999a). The IRR is

inherent to a given project and not necessarily is available for the investor during

the life of the project.

THE OVERALL RATES OF RETURN

Shull refers to the overall rates of return as the rate of return of a terminal

value resulting from the intermediate cash flows compounded at the rate of

discount and an investment base defined as the present value of the amounts

invested through the life of the project. He distinguishes two overall internal rates

of returns: the first one, the Modified Internal Rate of Return (what he calls the

MIRR method) that implies that any positive cash flow followed by a negative cashflow, will be used to fund the negative cash flow. In this case, the terminal value

is the future value of all positive net cash flows after funding for any negative

cash flow that occurs in the future (this means that positive cash flow can not

fund past negative cash flows) and the project investment base is the present

value of any negative net cash flow, after being fundedby the positive cash flows.

The second overall rate of return (what he calls the IRR* method) considers as

terminal value the future value of the net positive cash flows (without fundingthe

intermediate negative cash flows) compounded at the discount rate and the

investment project base is the present value of all negative net cash flow (without

any fundingfrom the positive net cash flows). In the first case the interpretation

of the MIRR is the yield produced by the net investment of external funds. In the

second case the interpretation of the IRR* is the yield produced by the total project

investment. Shull advocates for the MIRR as the correct one. For consistency with

the idea of measuring the benefits and value generated by the project (see Velez-

Pareja, 1999a), in this paper the IRR* will be considered the correct one.

In general, an overall rate of return is defined by

1

1

=

n

IB

TVORR (2)

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where ORR is the overall rate of return, TVis the terminal value and is the

amount compounded of the net positive cash flows at the discount rate, IBis the

investment base and is the present value of all negative net cash flow and nis the

horizon period for the project.

THE WEIGHTED INTERNAL RATE OF RETURN (WIRR)

How to include the first assumption? Just make explicit the reinvestment

assumption in the cash flow and recalculate the IRR as indicated by theORR, above.

This is, all the inflows will be compounded at the rate of discount up to period n and

all the outflows will be discounted at the discount rate to instant 0. This will convert

the original cash flow of the project into a cash flow with an investment outlay atinstant 0 and an inflow at period n. This cash flow defines a modified rate of return

(MIRR or IRR* defined above or the internal rate of return with reinvestment IRRR).

How to include the second assumption? Just find the difference between the

present value of all the outflows just calculated in the previous paragraph and the

maximum outflow in the collection of alternatives. This figure will be compounded

up to period n at the discount rate. This cash flow (an outflow at instant 0 of the

difference and an inflow of that difference compounded at the discount rate) defines

an IRR equal to the rate of discount.

The sum of the two cash flows will define an overall rate of return. This overall rate

of return is the adjusted rate of return ( ADJORR in Shulls notation) or weighted

internal rate of return (WIRR).

Then, the new rate of return is

( )1

1)(

1

++==

n

MAX

n

AMAXAAA

IB

iIBIBTVWIRRADJORR (3)

where ADJORRA is the adjusted rate of return proposed by Shull, WIRRA is the

weighted internal rate of return , TVAis the compounded value at the discount rate

of inflows, IBMAXis the maximum present value of the amount invested in each

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alternative and IBAis the amount invested in the alternative under study, nis the

number of periods and iis the discount rate.

The proof that WIRR or ADJORR are consistent with accept/reject decisions with

NPV and its ranking is found in Shull (1992). However, a simpler proof that they

are consistent with NPV is provided.

Let project A has a larger investment amount than project B, this is

BA IBIB (4)

Then

( )1

1)(

1

++==

n

A

n

BABBB

IB

iIBIBTVWIRRADJMIRR (5)

and

( )11

1)(11

=

++==

n

A

An

A

n

AAAAA

IB

TV

IB

iIBIBTVWIRRADJMIRR (6)

If

BBAA WIRRADJMIRRWIRRADJMIRR === (7)

( ) n

A

n

BABn

A

A

IB

iIBIBTV

IB

TV

11

1)(

++=

(8)

and

( ) ( ))(

11 BAnB

n

A IBIBi

TV

i

TV

++=+(9)

and

( ) ( ) ( )BBn

B

ABAn

B

AAn

A NPVIBi

TVIBIBIB

i

TVNPVIB

i

TV=

+=+

+==

+ 1)(

11(10)

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This proof is also valid for the case when the investment in B is larger than the

investment in A. From this proof the transitivity of the ADJORR or the WIRR can be

deducted.

EXAMPLE 1

Consider four mutually exclusive alternatives and a discount rate of 6% per

annum. (This is the same example presented by Shull, 1992).

Time A B C D

0 -10 -10 -10 -20

1 5 9 0 0

2 -9 -5.0 0 03 25 25 33.7 33.7

IRR 29.5% 58.2% 49.9% 19.0%

NPV $7.7 $15.0 $18.3 $8.3

In this case a contradiction in the optimal selection of alternatives is found

between NPV and IRR . Alternative C has the highest NPV, but alternative B has the

highest IRR. The NPV ranking is C>B>D>A. The IRR ranking is B>C>A>D.

According with Shull the ADJORR will be calculated with MIRR.

Time A B C D

0 -10.0 -10.0 -10 -20

1 0 0 0 0

2 -3.7 4.5 0 0

3 25.0 25.0 33.7 33.7

Observe that cash flow at time 1 funds outlay at time 2, assuming it is

compounded one period at 6%. Now the negative cash flow at period 2 is

discounted to instant 0 and any positive intermediate cash flow is compounded to

the last period.

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Time A D-A TOTAL A B D-B TOTAL B C D-C TOTAL C D

0 -13.3 -6.7 -20 -10.0 -10 -20 -10 -10 -20 -20

1 0 0 0 0 0 0 0 0 0 0

2 0 0 0 0 0 0 0 0 0 0

3 25.0 8.0 33.0 29.8 11.9 41.7 33.7 11.9 45.6 33.7

ORR (MIRR) 23.4% 6.0% 43.9% 6.0% 49.9% 6.0% 19.0%

ADJORR 18.2% 27.78% 31.63% 19.0%

NPV $7.7 $15.0 $18.3 $8.3

Observe that the NPV for the four alternatives is the same as the calculated with

the original cash flows. The ranking with NPV and ADJORR is the same: C>B>D>A.

According to the proposed method the WIRR will be calculated with the IRR* or

IRRWR.The IRRWR is the rate of return resulting from the cash flow resulting from

compounding all the positive cash flows up to the last period and discounting allthe negative cash flow to the instant 0.

Time A D-A TOTAL A B D-B TOTAL B C D-C TOTAL C D

0 -18.0 -2.0 -20.0 -14.4 -5.6 -20.0 -10 -10 -20 -20

1 0 0 0 0 0 0 0 0 0 0

2 0 0 0 0.00 0 0.00 0 0 0 0

3 30.6 2.4 33.0 35.1 6.6 41.7 33.7 11.9 45.6 33.7

ORR (IRR*/IRRWR) 19.3% 6.0% 34.4% 6.0% 49.9% 6.0% 19.0%

WIRR 18.2% 27.78% 31.63% 19.0%

NPV $7.7 $15.0 $18.3 $8.3

Observe that the WIRR and the ADJORR give the same results. Also note that the

NPV of each alternative is the same as the calculated with the original cash flow.

The rankings with NPV and ADJORR and WIRR are the same: C>B>D>A. The

ranking in both cases coincides with the NPV ranking.

The ORR for both methods differs because in the second case the cash flow

refers to the free cash flow and in the first case refers to the cash flow of external

financing. Observe that the inconsistency in the optimal selection (not in theranking) -for this example- could be solved just calculating the ORR s.

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In summary, the ranking by the different methods is:

Ranking IRR*/IRRWR

Ranking WIRR/ADJORR

Ranking ORR(MIRR)

Ranking NPV

I ( C ) 49.9% I ( C ) 34.6% I ( C ) 49.9% I ( C ) 18.3II ( B ) 34.4% II ( B ) 27.8% II ( B ) 43.9% II ( B ) 15.0III ( A ) 19.3% III ( D ) 19.0% III ( A ) 23.4% III ( D ) 8.3IV ( D ) 19% IV ( A ) 18.2% IV ( D ) 19.0% IV ( A ) 7.7

Observe that WIRR and ADJORR coincides with the NPV ranking, and hence with

the optimal selection.

EXAMPLE 2

This example was presented by Fleischer (1966). An investor has a discount

rate of 5% per annum and is analyzing two mutually exclusive alternatives A and

B with the following information:

Alternative Investment $ Annual

benefits $

Horizon Salvage value

at n $

A 20,000 3,116 10 years 0

B 10,000 1,628 10 years 0

Available capital $20,000

NPVA(5%) = $4,060,95 and IRRA= 9.00%

NPVB(5%) = $2,570,98 and IRRB =10.01%

In this example a definite contradiction between the NPV, IRR and B/CR is found.

Alternative Investment $ Terminal value at n $ ORR (IRRWR)

A 20,000 39,192.7 6.96%

B 10,000 20,476.8 7.43%

Observe in this case that the reinvestment assumption is not enough to solve

the contradiction. When the difference in the invested amount is considered,

then,

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Alternative Investment $ Terminal value at n $ ORR (IRRWR)

A 20,000 39,192.7 6.96%

B 10,000 20,476.8 7.43%

B-A 10,000 16,288.9 6.00%

Total B 20,000 36,765.8 6.28%

Alternative Investment $ Terminal value at n $ ADJORR OR WIRR

A 20,000 39,192.7 6.96%

Total B 20,000 36,765.8 6.28%

Now the inconsistency is solved.

As a by product, this method eliminates the problem of multiple rates of

returns.

THE EXPANDED BENEFITCOSTRATIO (EB/CR)

When analyzing mutually exclusive alternatives it is common use to calculate

the benefit cost ratio or profitability index. As mentioned above, this indicator is a

relative one and does not take into account the amounts of the investment (more

precisely, the differences between the amounts invested). The profitability index

might generate inconsistencies in the rankings as compared with NPV rankings,

as well. As was done with the IRR, the implicit assumptions found in the NPV

model have to be included in B/CR calculations.

In this case it is necessary to include just the assumption regarding the

differences in amount invested, because when discounting the cash flow at the

discount rate, the reinvestment at that rate is implicitly assumed in the

calculation.By definition, the B/CR considers the present value of inflows divided by the

present value of outflows. The only adjustment to do is to include the present

value of the inflow and the outflow of the additional cash flow. This is, the cash

flow composed by the difference of the present values of the amounts invested

(IBA-IBB, in the case of investments A and B and A>B) and present value of the

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compounded value of this difference. The present value of the inflow and the

outflow is just the difference (IBA-IBB) calculated for instant 0. Given this

definition, the EB/CR is

)(

)(PV

EB/CRA

Ainflows

A

AMAX

AMAX

IBIBPV

IBIB

++

= outflows (11)

where EB/CRA is the expanded benefit cost ratio, PVis the present value at the

discount rate of the absolute value of inflows or outflows, IBMAXis the maximum

present value of the amount invested in each alternative and IBAis the present

value of the amount invested (out flows) in the alternative under study.

The proof that EB/CRis consistent with accept/reject decisions and with NPV

ranking is simple.

Let

1)(

)(PVEB/CR

A

Ainflows

A =++

=AMAX

AMAX

IBIBPV

IBIB

outflows

(12)

then,

)()(PV AAinflows AMAXAMAX IBIBPVIBIB +=+ outflows (13)

and

0PVAAinflows

== ANPVPVoutflows (14)

On the other hand, for proofing the consistency with NPV rankings, if

)(

)(PV/

)(

)(PVEB/CR

B

Binflows

A

Ainflows

A

BMAX

BMAX

B

AMAX

AMAX

IBIBPV

IBIBCREB

IBIBPV

IBIB

++

==++

=outflowsoutflows

(15)

but,

MAXBMAXAMAXIBIBIBPVIBIBPV =+=+ )()( BA outflowsoutflows (16)

then,

MAX

BMAX

MAX

AMAX

IB

IBIB

IB

IBIB )(PV)(PVBinflowsAinflows

+=

+(17)

and

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BBAA NPVIBNPVIB === BinflowsAinflows PVPV (18)

EXAMPLE 3

Working with the same example 1,

Time A B C D

0 -10 -10 -10 -20

1 5 9 0 0

2 -9 -5.0 0 0

3 25 25 33.7 33.7

NPV $7.7 $15.0 $18.3 $8.3

B/CR 1.43 2.0 2.8 1.41

In this case there is not a contradiction between NPV and B/CR when selecting

the optimal alternative. However, the NPV ranking is C>B>D>A, while the B/CR

ranking is C>B>A>D.

Time A D-A TOTAL A B D-B TOTAL B C D-C TOTAL C D

0 -18.0 -2.0 -20.0 -14.4 -5.6 -20.0 -10 -10 -20 -20

1 0 0 0 0 0 0 0 0 0 0

2 0 0 0 0.00 0 0.00 0 0 0 0

3 30.6 2.4 33.0 35.1 6.6 41.7 33.7 11.9 45.6 33.7

19.3% 6.0% 34.4% 6.0% 49.9% 6.0%

NPV $7.7 $15.0 $18.3 $8.3

EB/CR 1.38 1.75 1.91 1.41

The calculations are, for alternative A:

Present value of inflows for A= PV inflows A=5/(1+6%)+25/(1+6%)3 = 25.71

Present value of extra amount invested IBA= PVoutflows A = 10 + 9/(1+6%)2 = 18.0

Maximum amount invested IBMAX = 20IBMAXIBA = 2.0

38.10.20

0.27.52EB/CR

A =+

=

The ranking by EB/CR is C>B>D>A and coincides with the NPV ranking.

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A summary for example 3 and according to the summary presented for example

1, follows:

Ranking B/CRIRR*/IRRWR basis

Ranking EB/CRWIRR/ADJORR basis

Ranking B/CRORR(MIRR) basis

Ranking NPV

I ( C ) 2.83 I ( C ) 1.91 I ( C ) 2.83 I ( C ) 18.3

II ( B ) 2.04 II ( B ) 1.75 II ( B ) 2.50 II ( B ) 15.0III ( A ) 1.43 III ( D ) 1.41 III ( A ) 1.58 III ( D ) 8.3IV ( D ) 1.41 IV ( A ) 1.38 IV ( D ) 1.41 IV ( A ) 7.7

As can be seen, the EB/CR and the B/CR with the cash flows from example 1,

maintain the same rankings as before. In particular, the EB/CR is consistent with

the NPV ranking.

EXAMPLE 4

Analyzing the same situation as in example 2, then NPVA(5%) = $4,060,95 andB/CR A= 1.203 and NPVB(5%) = $2,570,98 and B/CRB= 1.257.

In this case the contradiction can be solved by the EB/CR, as follows:

203.1000,20

95.060,24/ ==ACREB

129.1000,20

000,1098.570,12/ =

+=BCREB

As EB/CRA>EB/CRB, then, A is preferred to B, as recommended by the NPV rule.

Ochoa (1987) conducted a simulation to test this indicator and there was 100%

coincidence with NPV rankings.

THE IMPORTANCE OF RELATIVE MEASURES

Someone might ask with reason, if academics know that NPV is superior to IRR

and B/CR, why expending so much effort and time working out solutions like

those presented above? The reason is simple: relative measures such as IRR and

B/CR are widely used and they have an intuitive appeal that apparently makes

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them appear as more ease to understand. More, some teachers and textbooks on

project appraisal suggest strongly the calculation of the NPVandthe IRR andthe

B/CR. When those teachers are asked why to teach the student to work with the

three methods, the answer is just to be sure that the decision is the correct one

(!). Imagine what could happen when an analyst presents a set of mutuallyexclusive alternatives with the NPV and the IRR and the B/CR and the

recommendation is to select the project with the highest NPV, but with a non-

optimal IRR or B/CR. Probably more than one of the directors will ask why to

choose an alternative that does not yield the highest return. Probably the analyst

will be fired! The WIRR and the EB/CR could be a good option to avoid

embarrassing situations like the above-mentioned.

SUMMARY

In this paper a particular example of the process to arrive to a piece of

knowledge has been presented. The advances and improvements, successes and

mistakes occurred during the process are shown as a pedagogical contribution to

the work many students develop in doing their research work. Two alternative

methods for relative indicators ( IRR and B/CR) that present inconsistencies with

the NPV ranking of alternatives when mutually exclusive projects are studied. The

first is the adjusted overall rate of return, ADJORR presented by Shull (1992) orweighted internal rate of return, WIRR presented by Velez-Pareja (1979a, 1979b,

1979c and 1994). The second one is the expanded benefit cost ratio or prof itability

index, presented by Velez-Pareja (1979a, 1979b, 1979c and 1994). Both methods,

WIRR and EB/CR are consistent with NPV accept/reject criteria and NPVranking. With

WIRR the problem of multiple IRRs is overcome.

Methods like these are alternatives for those that prefer the project analysis and

appraisal based on IRR or B/CR. For an apology of the IRR , see Martin7 (1995).

BIBLIOGRAPHIC REFERENCES

BACON, P.W., 1977, "The Evaluation of Mutually Exclusive Investments". Financial

7Independent Contractor & Northwest Vista College, San Antonio

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Management, Vol. 6 No., Summer, pp 55-58.

BEAVES, ROBERTG., 1988, Net Present Value and Rate of Return: Implicit and

Explicit Reinvestment assumptions, The Engineering Economist, Vol 33.

Summer, pp. 275-302

BEIDLEMAN, C.R., 1984, Discounted Cash Flow Reinvestment Rate Assumptions,The Engineering Economist, Vol. 29. #2, pp127-137.

BERNHARD, RICHARD H., 1989, Base Selection for Modified Rates of Return and its

Irrelevance for Optimal Project Choice, The Engineering Economist, Vol 35.

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CANADA, J.R. AND WHITE, JR. J.A., 1980, Capital Investment Decision Analysis for

Management and Engineering, Prentice Hall Inc., Englewood Cliffs, N.J.

ECHEVERRI , E., 1987, Modelo aleatorio para ordenar alternativas de inversin.

Monografa de investigacin. Magister en Administracin (MBA),

(unpublished), Universidad de los Andes, Bogot.

FLEISCHER , G.A., 1966, "Two Major Issues Associated with the Rate Return Method

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GRANT, E.L. AND IRESON, W.G, 1960, Principles of Engineering Economy. 4th ed. The

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LIN , STEVEN A. 1976, The Modified Internal Rate of Return and Investment

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OAKFORD, R.V., BAIMJEE, S.A. ANDJUCKER, J.V., 1977, "The Internal Rate of Return

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OCHOA, L. F., 1987, Dos casos sobre Evaluacin de Proyectos, Monografa de

Investigacin, Magister en Administracin (MBA), (unpublished), Universidad

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SHULL, DAVID M. 1992, "Efficient Capital Project Selection Through a Yield-Based

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