Issues in Accounting EducationVol. 13, No. 3August 1998

Capital Budgeting: Some Exceptionsto the Net Present Value Rule

Anil Arya, John C. Fellingham and Jonathan C. GloverABSTRACT: Textbooks tend to emphasize the net present value (NPV) rule,often arguing that it is theoretically superior to other methods. Yet other meth-ods, many of which do not involve discounting, are also used in practice. Hence,one of two conclusions can be drawn: (1) firms are making suboptimal deci-sions or (2) the assumptions underlying the NPV rule are not always met inpractice. The purpose of this paper is to present simple numerical exampleswherein applying the NPV rule leads to erroneous decisions. The exampleshighlight the assumptions underlying the NPV rule.

INTRODUCTION

Capital budgeting is a vital ac-tivity. It is the process bywhich organizations make

long-term investment decisions.Textbooks in accounting and fi-nance discuss numerous evaluationcriteria, including payback period,accounting rate of return, internalrate of return, and Net PresentValue (NPV).i These criteria canlead to differing conclusions.

The NPV rule of "accepting aproject if and only if its NPV is posi-tive" is based on the intuitive premisethat money today is worth more thanthe same amount of money in the fu-ture. Textbooks tend to emphasizethe NPV rule, often arguing that it istheoretically superior to other meth-ods (see, for example, Kaplan andAtkinson 1989, 474-475; Zimmerman1995,119). Yet other methods, manyof which do not involve discounting,are also used in practice. For ex-ample, in a survey referred to inHorngren et al. (1997, 794), morefirms reported using the payback

method either as a primary or sec-ondary criterion to evaluate projectsthan any other method.^

Since companies use these othermethods, one of two conclusions canbe drawn: (1) firms are maMng sub-optimal decisions or (2) the assump-tions underlying the NPV rule are not

' See, for example, Horngren et al. (1997),Kaplan and Atkinson (1989), Ross et al.(1995) and Zimmerman (1995).

2 Another survey reported that the paybackmethod is commonly used as a secondary cri-terion, but not as a primary criterion (seeRoss etal. 1995,219).

Anil Arya is an Associate Professor andJohn C. Fellingham is a Professor atOhio State University and JonathanC. Glover is an Associate Professor atCarnegie Mellon University.

We thank Doug Schroeder, Li Zhang, stu-dents at Carnegie Mellon University andOhio State University, Wanda Wallace(the editor), and two anonymous refer-ees for helpful comments. Anil Arya ac-knowledges support from Ernst &Young. John Fellingham acknowledgessupport from theH. P. Wolfe Foundation.

500 Issues in Accounting Education

always met in practice. The purposeof this paper is to present numericalexamples wherein applying the NPVnale leads to erroneous decisions. Theexamples highlight the assumptionsunderlying the NPV rule.

Although the simplest version ofthe NPV rule deals with the case inwhich cash flows are known with cer-tainty, uncertainty in cash flows canbe incorporated by taking expecta-tions over cash flows and discount-ing using a risk-adjusted interestrate. However, uncertainty is relatedto two assumptions underlying theNPV rule that are not so easily dealtwith. The NPV rule assumes that (1)the project approval decision is a"now-or-never" decision (if a projectis turned down now it cannot be un-dertaken in the future) and (2) deci-sions are made either in a single-per-son firm or in a multi-person firm inwhich there are no information asym-metries among individuals.^ Our ex-amples deal with cases in which theseassumptions do not hold.

The examples are intentionallysimple. They are intended to providethe reader with an appreciation formanagement's use of a variety of cri-teria and an understanding of someof the underlying considerations. Inpractice, uncertainty, informationasymmetry problems, and multi-period, multi-project considerationsgreatly complicate capital budgeting,beyond the focus of this paper.

When the NPV rule's assumptionsare violated, the use of multiple cri-teria is a way of evaluating the projectfrom different perspectives. If manyof the criteria suggest the projectshould be taken, the chance is greaterthat the project is desirable. AsDemski (1994, 385) writes, there is"ambiguity in the present value frameitself... .In this case, we then acknowl-

edge an ambiguous framing exercisecoupled with a portfolio of ap-proaches to the framing task." Rosset al. (1995, 218-219) present a simi-lar view: "[b]ecause the true NPV isunknown, the astute financial man-ager seeks clues to assess whetherthe estimated NPV is reliable. Forthis reason, firms would typically usemultiple criteria for evaluating aproject... [if] different indicators seemto agree [then] it's 'all systems go.'"

The remainder of the paper is or-ganized into three sections. The sec-ond section relaxes the now-or-neverassumption in order to study the op-tion value of waiting. The third sectionrelaxes the "no information asymme-tries" assumption and, in particular,studies decentralized information.The fourth section studies the effectof reducing information asymmetries.

OPTION VALUE OF WAITINGTextbooks suggest an exception to

the NPV rule: If two mutually exclu-sive projects are being considered,then only the project with the higherNPV is to be accepted, even if bothprojects have positive NPVs (see, forexample, Brealey and Myers 1996, 97-100). Two projects are mutually exclu-sive if undertaking one precludes un-dertaking the other. For example, if afirm owns a tract of land and has achoice of building either a warehouseor a plant on the land, then these twoprojects are mutually exclusive.

The issue of mutually exclusiveprojects is traditionally discussed inthe context of multiple projects. More

For analyses in which the first assumption isrelaxed, see, for example, Balakrishnan andBhattacharya (1997), Dixit and Pindyck(1994),Ross (1995) andTrigeorgis (1996). Foranalyses in which the second assumption isrelaxed, see, for example, Antle and Eppen(1985) and Harris et al. (1982).

Arya, Fellingham and Glover 501

recently, the evaluation of mutuallyexclusive projects has been extendedto the case of one project—a projectmay be in competition with itselftaken at a later date. Often, by turn-ing a project down today, the firm pre-serves the option to invest in theproject at a later date. By investingtoday, this option is lost. When is keep-ing the investment option alive betterthan investing in the project today?^

The Resolution of UncertaintyOne advantage of waiting to in-

vest in a project is that the firm mayreceive better information about theproject's cash fiows, i.e., uncertaintymay be reduced over time. Considerthe following example. If a project isundertaken, the required cash out-flow (investment/cost) today is 100.The project generates a cash inflowof 120 or 90, each equally likely, in oneyear from now. For simplicity, as-sume the discount rate is 0 through-out the paper. Alternatively, one canview the cash flow numbers as beingin today's dollars (they are alreadydiscounted). The project's expectedNPV is .5(120 - 100) + .5(90 - 100) = 5,and the NPV rule would lead the firmto accept the project.

Suppose that by rejecting theproject this year and waiting until nextyear to make a final Investment deci-sion, the firm leams exactly what theproject's cash inflow will be. If the firmwaits, it will invest in the project if andonly if it leams the cash infiow is go-ing to be 120. The project's expectedNPV is .5(120 - 100) + .5(0) = 10. Be-cause of the reduction in uncertainty,it is better to wait to make the invest-ment decision.

The Luck of the DrawEven if uncertainty is not re-

duced, it may be worthwhile to wait

to accept a project. By waiting, it ispossible the project will be under-taken under more favorable circum-stances (e.g., interest rates may de-cline). Consider the following ex-ample. If a project is undertaken, itgenerates a cash inflow of 100. Thecost is 60 or 90, each equally likely.When the project is undertaken, theproject's exact cost is known.Whether the cost is 60 or 90, theNPV rule dictates the project be ac-cepted, since its NPV is either 40 or10.

If the project approval decisionis delayed, the cost depends on theprevailing economic conditions.Suppose the cost in the second pe-riod is again equally likely to be 60or 90, and costs are uncorrelatedacross periods. Returning to theinitial project approval decision, itis optimal to reject the project if thecost is 90; by waiting, the environ-ment either stays the same or im-proves. If the cost is 90, the ex-pected NPV of accepting the projecttoday is 100 - 90 = 10 but by waitingis .5(100 - 60) + .5(100 - 90) = 25.Waiting enables the firm to obtaina new, potentially more favorableenvironment in which to undertakethe project.

Why does the NPV rule not yieldthe correct answer in the above ex-amples? The NPV rule implicitly as-sumes the project choice is a now-or-never decision. In both examples, weassumed the firm's opportunity toinvest in a project is not lost if theinvestment decision is delayed, i.e.,the project choice is a "now-or-later" decision.

Our discussion of the option value perspec-tive to capital budgeting is based on Dixitand Pindyck (1994) and Ross (1995).

502 Issues in Accounting Education

DECENTRALIZEDE^ORMATION AND

CONTROLUp to this point, we have had a

simple picture of a firm in mind. Thefirm is owned and operated by a singleindividual. In this environment, capi-tal budgeting can be modeled as a de-cision problem. In a multi-person firm,capital budgeting is perhaps bettermodeled as a control problem. In or-der to control managers' behavior(e.g., limit "budget padding"), a bud-get center may find it optimal to com-mit to project approval/rejection de-cisions that would not be optimal inthe single-person world of decisiontheory. Key ingredients of controlproblems are conflicts of interest andinformation asymmetries among firmparticipants.^

We next study capital budgetingas a control problem.^ Consider a firmwhose participants consist of a cen-ter responsible for a project approvaldecision and a division manager whoimplements any approved project.The cash inflow from the project is100, and the cost is either 60 or 90,each equally likely. The project is anow-or-never project. Since theproject's NPV is always positive, theNPV rule dictates the project be ac-cepted. However, as we will show,this is not the correct rule to applyin the presence of a control problem.

Because of his proximity to op-erations, the division manager learnsthe exact cost of the project beforethe project approval decision ismade. The center knows only the dis-tribution over costs. Hence, there isan information asymmetry. In par-ticular, information is decentralizedin that the manager has an informa-tional advantage over the budgetcenter. Divisions submit detailedcost budgets to the center prior to

approval. We refer to the amount ofreport-contingent resources pro-vided by the center to the manageras the project's funding. If the divi-sion manager always reports truth-fully, it is optimal to approve theproject and provide funding as re-quested by the manager. Theproject's expected NPV is .5(100-60)+ .5(100-90) = 25.

However, one problem with de-centralized information is that divi-sion managers often show a markedtendency to pad their budgets. Slackin the budgets makes life more pleas-ant in the divisions. If a divisionalbudget is successfully padded, divi-sional personnel need not work ashard or as efficiently. Also, they maybe able to consume corporate prereq-uisites. The center, on the otherhand, would like to minimize theamount of slack in budgets (otherthings being equal).

The center supplies all funds forcapital expenditures and receives thecash inflow (if the project is under-taken). The division manager bearsthe cost of production and consumesany slack. If the center supplies thefunds requested by the manager, themanager will always report the costof the project is 90 (even when it is60). The center's expected profit is100 - 90 = 10, and the manager's ex-pected slack is .5(90-60) + .5(90 - 90)= 15.

Accounting and control problems are closelylinked. Accounting is based fundamentally onstewardship relationships, whereby re-sources are entrusted to individuals whohave to account for the use of those re-sources. It seems unlikely the incentives ofthe accountor (steward) would completelycoincide with those of the accountee (Ijid1975,35).The discussion of capital budgeting as a con-trol problem is based on Antle and Eppen(1985).

Asya, Fellingham and Glover 503

Can the center earn higher ex-pected profits? Consider the follow-ing contract, which we refer to as the"Slack Contract." The center com-mits to accepting the project and pro-viding funding of 90, irrespective ofthe manager's cost report. Under thiscontract, the manager is indifferentbetween reporting truthfully andpadding his budget. When indifferent,we assume the manager will reporttruthfully.'' The Slack Contract isequivalent to funding all projects atthe level requested by the managerin the sense that, under both con-tracts, the same outcomes occur—the project is always accepted andfunding of 90 is provided. The equiva-lence of these two contracts is a spe-cial case of what is referred to as the"Revelation Principle."

The Revelation Principle statesthat any equilibrium outcome of anymechanism can be replicated as atruth-teUing equilibrium of a revela-tion mechanism. Two words deserveclarification. By an "equilibrium," wemean the manager's reporting strat-egy is a best response to the contract.By a "revelation mechanism," wemean a mechanism under which themanager is asked to report his privateinformation. The Revelation Prin-ciple is a useful tool in economics be-cause it greatly simplifies the task offinding an optimal mechanism. It jus-tifies restricting attention to mecha-nisms under which the manager'sown best interest is to truthfully re-veal his private information.

There is another contract that in-duces truthful revelation of informa-tion—the center accepts the project ifand only if the manager reports theproject's cost is 60. We refer to thiscontract as the "Rationing Contract,"since the center commits to rejectinga project when the cost is 90, even

though it is guaranteed to generate acash infiow of 100. Under this contract,when the cost is 60, the manager'sslack is 0 whether or not he pads hisbudget; hence, the manager reportstruthfully. The center's expected profitis .5(100 - 60) = 20, and the manager'sexpected slack is 0. Since the Slackand Rationing Contracts are the onlyones that induce truthful revelation ofinformation (other than always reject-ing the project) and the RevelationPrinciple holds, we can conclude thatthe Rationing Contract is optimal inour example.̂ As an optimal responseto the manager's desire to pad his bud-get, the center commits to rejectingsome positive NPV projects. Rationingdisciplines the manager's reporting.

The benefit of the Rationing Con-tract relative to the Slack Contractis that when the cost is 60, the cen-ter ends up paying 60 instead of 90.The saving in managerial slack is.5(90 - 60) = 15. The cost of the Ra-tioning Contract relative to the SlackContract is that when the cost is 90,a positive NPV project is forgone.The cost of forgone production is.5(100 - 90) = 5. The benefit exceedsthe cost by 15 - 5 = 10; recall thatthis is the difference between thecenter's expected profit under theRationing Contract and under theSlack Contract (20 - 10 = 10) .̂' The manager is not a pathological liar; he lies

only if it increases his slack.* We have considered a simple accept/reject

decision—the project is either accepted andfully funded or rejected. Even if it is possibleto accept some fraction of the entire projectand the cost and revenue functions are lin-ear, either the Slack or the Rationing Con-tract continues to be optimal. We formulatethis problem as a linear program and presentits solution in the appendix.

^ If the probability of the low cost is .25, the ben-efit of rationing is .25(90 - 60) = 7.5 and thecost of rationing is .75(100 - 90) = 7.5. If theprobability is even lower, the Slack Contractis strictly preferred to the Rationing Contract.

504 Issues in Accounting Education

The glue that holds these con-tracts (and the Revelation Principlein general) together is commitment.The center commits to less aggres-sive use of information in order toelicit that information. Under theSlack Contract, it is important themanager believes funding of 90 will beprovided even when he budgets 60.Under the Rationing Contract, it isimportant the manager believes thatif he budgets a cost of 90, the centerwill reject the project. The role ofcommitment is particularly impor-tant under the Rationing Contractsince, when the manager reports thecost is 90, both parties would be will-ing to tear up^the contract and writea new one under which the 90 costproject is accepted. Doing so makesboth parties better off (when the costis 90). However, if the manager an-ticipates the contract will be renego-tiated when a cost report of 90 is sub-mitted, he will pad his budget whenthe cost is 60, making the centerworse off. Hence, it is valuable for thecenter to build a reputation as some-one who sticks to her agreements.

The Slack and Rationing Con-tracts are hurdle rate contracts. Thecenter commits to accepting theproject if its cost is equal to or belowsome cutoff and rejecting the projectif its cost is above that cutoff. If theproject is accepted, the funding pro-vided to the manager is the cutoffcost. The optimal contract is also ahurdle rate contract when there aremore than two costs.

REDUCING THEINFORMATION ASYMMETRYNonstrategic Information Sources

Suppose the center can acquireinformation about the cost of theproject, which reduces the informa-tion asymmetry between the center

and the manager.^" The source ofthe information is unmodeled; fornow, treat it as a black box. Howmuch is the center willing to pay forthis information? Return to the ex-ample presented in the previoussection. If perfect information canbe acquired by the center, theproject is always accepted and thefunding is equal to the cost; thecenter's expected profit is 25. (Notethat in this setting the NPV ruleapplies.) In the absence of the in-formation system, the optimal con-tract is the Rationing Contract,which provides the center with anexpected profit of 20. Hence, thevalue of perfect information is 5.

In the example, there is rationingin the presence of an informationasymmetry and no rationing in its ab-sence. This might lead one to conjec-ture that a reduction in informationasymmetry is always accompanied bya reduction in rationing. We nextshow that when the center installs aninformation system that reduces theinformation asymmetry, she some-times optimally increases the amountof rationing. In other words, reduc-ing the information asymmetry doesnot necessarily move us closer to theNPV rule.

Suppose there are now four pos-sible costs, 60, 70, 80 and 90, and theprobability of each cost is .2, .05, .05and .7, respectively. As before, thecash infiow is 100. First, suppose thecenter knows only the above prob-abilities, while the manager knowsthe cost. In order to identify the op-timal contract, there are four hurdlerate contracts to compare. The opti-mal contract is to accept the project

Our discussion of the value of nonstrategicinformation is based on Antle andFellingham (1995).

Arya, FeUingham and Glover 505

no matter what cost the manager re-ports and to provide funding of 90.The center's expected profit is 10,and the manager's expected slack is7.5.

Now, suppose the center can in-stall an information system that re-veals whether costs are low (60 or 70)or high (80 or 90). The problem canbe decomposed into two binary costproblems and each can be solved asin the previous section of the paper.The center uses Bayes Rule to up-date her beliefs when she observesthe cost is low or high. For example,if she observes the cost is low, sheassigns probability .2/(.2 + .05) = .8to the cost being 60 and .05/(.2 + .05)= .2 to the cost being 70. The opti-mal contract is the Rationing Con-tract if the cost is low and the SlackContract if the cost is high. Thecenter's expected profit is 15.5, andthe manager's expected slack is .5.Hence, installing an information sys-tem that reduces the informationasymmetry is beneficial to the cen-ter. Perhaps surprisingly, it also re-sults in more rationing.

Intuitively, the cost is very likelyto be 90. Hence, rationing in this stateis very expensive. Without the infor-mation system, the center has to giveup production when the cost is 90 ifshe wants to do any rationing at all.This is so expensive that she opti-mally chooses not to ration at all.With the information system, whenthe center observes the cost is low,she can ration without the fear of giv-ing up production when the cost is90. As a result, the center finds it op-timal to ration when the cost is low.

In the examples considered so far,the center always prefers to decreasethe manager's informational advan-tage and the manager always prefersto increase his informational advan-

tage. The first conclusion is true aslong as the center has full powers ofcommitment, since she can alwayscommit to ignoring the information.However, the second conclusion isnot always true.

Continue with the previous ex-ample, except the probabilities of thecost being 60 and being 90 areswitched. That is, the costs are 60,70,80 and 90, and the probability of eachcost is .7, .05, .05 and .2, respectively.Without the information system, theoptimal contract is to accept theproject and provide funding of 60 ifthe reported cost is 60; otherwise, theproject is rejected. The center's ex-pected profit is 28, and the manager'sexpected slack is 0. With the informa-tion system, the optimal contract isthe Rationing Contract when the costis low and the Slack Contract whenthe cost is high. The center's ex-pected profit is 30.5, and themanager's expected slack is .5.Hence, the manager is actually bet-ter off when his informational advan-tage is reduced.

One way to think about the aboveresults is that the parties to a con-tract are affected not only by the sizeof the pie (expected production lessthe cost of production), but also byhow the pie is split. The way in whichthe pie is split depends on the size ofthe pie; production and distributionproblems do not decompose. Thecenter optimally chooses the size tomaximize the expected cash infiowless funding. With less of an informa-tion asymmetry, it is possible the cen-ter will choose to make the pie big-ger and, hence, both parties will bebetter off. In such situations, themanager has incentives not to sabo-tage (and, in fact, to help support) aninformation system that reduces hisinformational advantage.

506 Issues in Accounting Education

Strategic Information SourcesSuppose the information ob-

tained by the center is provided by astrategic source, i.e., an individualwho acts in his own self interest.^^Continue with the last example, ex-cept there are now two managers, Aand B, each proposing his ownproject. The center can accept one orboth projects. Costs in each divisionare correlated in that if manager i'scost is low (high), then manager j'scost is also low (high), i, j = A,B; J '̂ i-The center obtains informationabout manager i from both manageri's report and manager j's report.

The optimal contract depends onthe behavioral assumptions that areappropriate in predicting the way themanagers will play in the budgetinggame. One common behavioral as-sumption is due to John Nash. TheNash equilibrium concept assumesmanagers will play best responses toeach other's strategies. That is, givenmanager j 's strategy, manager ichooses a strategy that maximizeshis expected slack. If the center de-signs a contract that ensures onlythat truthful reporting is a Nash equi-librium, the optimal contract is asshown in figure 1.

The optimal contract is a rela-tive performance evaluation con-tract. The center will not accept ahigh budget from manager i unlessmanager j's budget is also high. Thecenter's expected profit is 2[.7(100- 60)-l-.05(0)+ .05(100 - 90)-l-.2 (100

- 90) ] = 61. This is exactly twice thecenter's profit under the nonstrate-gic information system.

Since the center's expected profit(per project) is the same under stra-tegic information supply as it is un-der nonstrategic information supply,it is tempting to conclude that thesource of the information is not im-portant. However, an additional con-trol problem is introduced with stra-tegic information. In our setting, asoften occurs with relative perfor-mance schemes, the optimal two-manager contract gives rise to animplicit collusion problem. Althoughboth managers reporting truthfully isa Nash equilibrium in the budgetinggame, so is each manager always re-porting his cost is 90. That is, thereexists an equilibrium under whicheach manager pads his budget to themaximum extent possible. Moreover,the padding equilibrium increaseseach manager's expected slack to22.5 (compared to .5 under the truth-telling equilibrium). This comes atthe center's expense; her expectedprofit is reduced to 20 under the pad-ding equilibrium. As Kreps (1990,702) points out, "if a mechanism ad-mits several Nash equilibria, some ofwhich are worse for the designer(and, more importantly, better forthe participants) than is the equi-librium that is desired, then one

" The discussion presented in this section isbased on Arya et al. (1996).

The Optimal Contract

Manager j Reports Manager i Reports i's Project is i's Funding60 or 70

80 or 90

6070, 80 or 90

60, 70, 80 or 90

AcceptedRejectedAccepted

600

90

Arya, Fellingham and Glover 507

worries that the participants will findtheir way to the wrong equilibrium.''̂ ^

The point of this example is two-fold. First, extracting informationfrom strategic sources can be a deli-cate exercise. Second, as additionalcontrol problems are introduced,capital budgeting decisions look lessand less like a direct application ofthe NPV rule.

This paper emphasizes that view-ing the NPV rule as governing allcapital budgeting decisions may notbe appropriate. The NPV rule appliesin limited settings in which there areno control problems and in which theprojects disappear if not undertakenimmediately. The NPV rule is not

wrong; given its underlying assump-tions, it is correct. The challenge isto learn how to make budgeting de-cisions in settings wherein the as-sumptions do not hold. In such set-tings, there is potentially a useful rolefor other capital budgeting tech-niques such as evaluations based onthe payback period, the internal rateof return, and the accounting rate ofreturn.

12 For an early paper on implicit collusion in aprincipal-multi-agent model, see DemsM andSappington (1984). For additional reading onthe implicit collusion problem, see Moore(1992) and Palfrey (1992). For reading on ex-plicit collusion (side contracting), see Tirole(1992).

APPENDIXDenote the cash inflow of the project by i (0 < i < I) and the cost of producing i by

ci (c = CL or c^, CL < Cjj < 1). The probability of c being CL is p^ and of c being Cjj is Pjj.(Since marginal cost is less than marginal revenue, the project always has a positiveNPV.) Let f denote the funding provided to the manager.

The contract the center offers the manager specifies cash inflow and fundingamounts as a function of the manager's cost report. The center maximizes the ex-pected cash inflow less funding, subject to the constraints that, under the contract:the manager has incentives to report truthfully (T), all funding comes firom the center(F), and the cash inflow is between 0 and I. The linear program that identifles theoptimal contract is presented below.

Max J

st

>>

The solution to this program is:1-c. 1-c.

i(CL) = I

i(Cjj) = I

f(CH) = CHI

i(CL) = I

i(Cjj) = 0

f(CH) = 0

508 Issues in Accounting Education

In the table on the prior page, the contract in the left colvimn is the Slack Contractand in the right column is the Rationing Contract.

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