Quantitative Finance, Vol. 9, No. 8, December 2009, 967–979

Investment decisions, net present value and

bounded rationality

CARLO ALBERTO MAGNI*

Department of Economics, University of Modena and Reggio Emilia, viale Berengario 51,

41100 Modena, Italy

(Received 2 April 2007; in final form 6 February 2009)

The Net Present Value maximizing model has a respectable ancestry and is considered by mostscholars to be a theoretically sound decision model. In real-life applications, decision makersuse the NPV rule, but apply a subjectively determined hurdle rate, as opposed to the ‘correct’opportunity cost of capital. According to a heuristics-and-biases-program approach, thisimplies that the hurdle-rate rule is a biased heuristic. This work shows that the hurdle-rate rulemay be interpreted as a fruitful strategy of bounded rationality, where several domain-specificand project-specific elements are integrated and condensed into an aspiration level. The paperalso addresses the issue of a productive cooperation between bounded and unboundedrationality.

Keywords: Corporate finance; Investments; Bounded rationality; Valuation; Cognition andeconomics

1. Introduction

This paper deals with the Net Present Value (NPV)methodology, a widely used tool for investment decisions.It is considered theoretically sound and normativelysuggested in many corporate finance textbooks (e.g.Copeland and Weston 1988, Bierman and Smidt 1992,Rao 1992, Damodaran 1999, Copeland et al. 2000,Brealey and Myers 2000, Fernandez 2002). The conceptof NPV is the brick of a normative building deep-rootedin the maximizing tradition of economics. The NPVmethodology may be summarized as follows: supposea decision maker has the opportunity of investing X0

in a one-period project generating a payoff X1. Leti¼X1/X0� 1 be the project’s rate of return and denoteby � the return rate of the next-best alternative availableto the investor; the project should be undertaken if andonly if

�X0 þX1

ð1þ �Þ4 0: ð1Þ

The left-hand side of (1) is called the project’s Net PresentValue. The NPV rule says that the decision maker shouldinvest in the project if its NPV is positive. The discount

rate � is the so-called opportunity cost of capital: if

a decision maker invests in the project she foregoes the

opportunity of earning � on her capital X0. The

opportunity cost of capital is then taken as a norm (in

the sense of Kahneman and Miller 1986) and represents

the return that the investor would earn if she invested

in the alternative course of action (see Buchanan 1969).

In general, if one faces two or more mutually exclusive

courses of action, one should calculate their NPVs by

discounting cash flows at the respective opportunity cost

of capital and choose the one with the highest NPV. The

NPV model traces back to Fisher (1930), whose analysis

was carried out under the assumption of certainty, and

may be formally derived from the following constrained

optimization problem, where the investor must select the

optimal consumption plan:

max uðc0, c1Þ, subject to ðm0 � X0 � c0Þð1þ �Þ

þm1 þ X1 ¼ c1,ð2Þ

where u(�) denotes the investor’s utility function, � is the

market interest rate and ct, mt, and Xt are, respectively,

the consumption, the income, and the investment flow at

time t¼ 1,2. Equation (2) represents the so-called Fisher

model. It provides for a separation theorem (the optimal

investment plan is the same regardless of the optimal*Email: magni@unimo.it

Quantitative FinanceISSN 1469–7688 print/ISSN 1469–7696 online � 2009 Taylor & Francis

http://www.informaworld.comDOI: 10.1080/14697680902849338

consumption bundle) and for a derivation of the net

present value rule as in equation (1) (see McMinn 2005).

Under uncertainty, one cannot say a priori that, forexample, an expected return of 100 is preferable to

an expected return of 90 if the former is riskier than the

latter: one is preferable in terms of return, the other is

preferable in terms of risk (risk-aversion is assumed).

So, if alternatives are not equivalent in risk, the NPV rule

may not be employed as such, because it only createsa partial ordering among alternatives. There is the need

of a theoretically correct cost of capital accounting for

risk. Among other models, portfolio theory (Markowitz

1952) and the Capital Asset Pricing Model (CAPM)

(Sharpe 1964, Mossin 1966) have been developed

in corporate finance for explaining the relation between

risk and return. The mathematical link between thesemodels and the NPV rule has been provided by several

scholars, who have proved, since the late sixties, that

maximization of shareholders wealth is equivalent

to maximization of NPV, where the cost of capital � in

equation (1) is the expected rate of return of a traded asset

equivalent in risk to the project under consideration

(equal systematic risk) and it is computed by making useof the CAPM security market line (e.g.Tuttle

and Litzenberger 1968, Hamada 1969, Bierman

and Hass 1973, Mossin 1969, Rubinstein 1973).A more recent version of the NPV rule is the ‘expanded

NPV’ (Trigeorgis 1986, Dixit and Pindyck 1994), which

makes use of arbitrage pricing or, alternatively, stochasticdynamic programming. It aims at taking into account

possible options intrinsic in the project (option to wait, to

abandon, to switch, to temporarily suspend, to enlarge

scale, etc.). Proponents of the expanded NPV often

call their approach the ‘real options approach’, which is

but a sophisticated version of the traditional NPV model,

where the set of alternatives is inclusive of the optionsimplicit in the project: ‘‘one can always redefine NPV by

subtracting from the conventional calculation the oppor-

tunity cost of exercising the option to invest, and then

say that the rule ‘invest if NPV is positive’ holds once this

correction has been made’’ (Dixit and Pindyck 1994, p. 7).

In other words, while the label ‘real options approach’ is

widespread in the literature, ‘‘if others prefer to continueto use ‘positive NPV’ terminology, that is fine as long

as they are careful to include all relevant option values

in their definition of NPV’’ (ibidem).yPast and recent empirical evidence shows that actual

decision makers turn the NPV model into a rule of

thumb: they use a subjective discount rate as opposed to

a rational cost of capital derived from some market model(e.g. CAPM, arbitrage theory, multifactor models).

As a result, they employ a criterion generated in the

realm of unbounded rationality but give it a distinctive

bounded-rationality flavor. This paper aims at showing

that this mixed rule may prove useful: the bounded/

unbounded dichotomy peters out and the hybrid rule

discloses a beneficial cooperation between bounded andunbounded rationality.

The paper is organized as follows. Section 2 surveyssome empirical evidence on the use of hurdle rates by real-life decision makers so that the NPV rule is interpretableas a bounded-rationality strategy. Section 3 presentsevidence that the hurdle-rate NPV rule may furnish close-to-optimal results when compared with the expandedNPV and that it may be viewed as a comprehensivemethodology that takes account of several domain-specific and project-specific variables. Section 4 showsthat the allegedly sound maximizing NPV have severalshortcomings, among which inconsistencies with acceptedstandards of rationality. Section 5 briefly hints ata normative role of bounded rationality and section 6presents some evidence indicating that the NPV rule ishardly interpretable as rigorously pertaining to either sideof rationality. Some remarks conclude the paper.

2. The hurdle-rate heuristic

There is some awareness in the corporate financeliterature that actual decision makers use the NPV rulebut do not use the cost of capital suggested by corporatefinance textbooks. They employ a subjective hurdle rate.Brigham (1975) surveyed 33 large, relatively sophisticatedfirms. Although 94% of them used DCF methodology,only 61% of the firms using DCF adopted the cost ofcapital as the discount rate. Summers (1987) surveyedcorporations on investment decision criteria, finding that94% of reporting firms use the NPV rule employinga discount rate independent of risk, which is suggestive ofthe use of hurdle rates. Dixit (1992) recognizes that ‘‘firmsinvest in projects that they expect to yield a return inexcess of a required or ‘hurdle’ rate’’ (p. 107). ‘‘Financescholars have always been puzzled by the durabilityof . . . the hurdle rate rule’’ (Ross 1995) and, in actual fact,‘‘we know that hurdle rates . . . are used in practice’’(McDonald 2000, p. 30); ‘‘it appears common for firms touse investment criteria that do not strictly implement theNPV criterion’’ (ibidem, p. 13), so that their ‘‘actionsdo not reflect the application of current financial theory’’(Gitman and Mercurio 1982, p. 29). Graham and Harvey(2002) surveyed 392 companies, and despite the fact thatmany companies claimed they do employ the NPVtechnique and use CAPM for estimating the cost ofcapital, they point out that the use of hurdle rates ispredominant. In particular, ‘‘small firms are significantlyless likely to use the NPV criterion or the capital assetpricing model and its variants’’ (p. 22). They find thatsometimes the use of hurdle rates is explicitly acknowl-edged: ‘‘small firms were inclined to use a cost of equitydetermined by ‘‘what investors tell us to require’’ [and a]majority (in fact, nearly 60%) of the companies said thatthey would use a single-company wide discount rate to

yA precursory example of the use of NPV for valuing real options may be found in Merrett and Sykes (1973, p. 129).

968 C. A. Magni

evaluate a new investment project, even though differentprojects are likely to have different risk characteristics’’(p. 12). Jagannathan and Meier (2002) (henceforth J&M)yobserve that ‘‘managers use a . . . hurdle rate’’ (p. 3)instead of employing a CAPM-derived cost of capital.Relying on Poterba and Summers (1995) analysis, theyfind that ‘‘hurdle rates are not . . . linked to the cost ofcapital’’ (p. 22). These findings suggest that actual choicebehaviors violate normative NPV and decision makersinterpret the rule as a satisficing strategy (Simon 1955),letting the discount rate be a predetermined cutoff levelwhich is considered satisfactory. Let i be the (expected)rate of return of the project, and let k be a hurdle ratesubjectively prefixed by the evaluator. A simple heuristicis the following:

undertake the project if i4 k,

otherwise search for another alternative until

the inequality is satisfied:

It is evident that such a rule may also be rephrased interms of NPV, with the hurdle rate k as the discount rate:given that X1¼X0(1þ i), the decision maker will acceptan investment with initial outlay X0 and (expected) finalreturn X1 if and only if�X0þX1/(1þ k)40. The invest-ment NPV is calculated with the rate k, which representsthe minimum rate of return acceptable by the investor.This rule is formally analogous to equation (1) but iscognitively different. Replacing � with k means shiftingfrom the idea of comparing equivalent-risk alternatives tothe idea of accepting a satisfactory rate of return. Both� and k are cutoff rates, but the former refers to anobjectively determined equivalent-risk alternative, thelatter refers to a subjectively determined aspiration level.So doing, one leaves the realm of unbounded rationalityand reaches the land of bounded rationality. Yet, bothrules are expression of an NPV criterion, relying as theyare on discounting cash flows at a prefixed discount rate.The investor invests in the project if the value ofher payoff function, calculated at the hurdle rate k, ispositive; that is, if the rate of return of the project exceedsthe aspiration level k. By adopting the hurdle-rateheuristic the evaluator does not confine herself toequivalent-risk alternatives. No alternative courseof action is called up, no comparison is accomplished,no maximization is involved. The rate k is not a rate ofreturn of an existing alternative, and therefore it is not anopportunity cost and does not represent a foregonereturn. It is a subjective threshold that identifiesthe personal minimum required rate of return from

the project. The heuristic-minded reasoner is uninterestedin knowing whether an alternative exists or not (eitherequivalent in risk or not) better than the one underexamination, she is just interested in reaching thataspiration level, for subjective reasons.

3. Does the NPV heuristic work?

McDonald (2000) contrasts the hurdle-rate rule with theexpanded NPV. The author aims at investigating‘‘whether the use of seemingly arbitrary investmentcriteria, such as hurdle rates and profitability indexes,can proxy for the use of more sophisticated real optionsvaluation’’ (p. 13). He wonders whether such ‘‘simplerules are relatively robust to changes in project character-istics’’ and whether ‘‘a single hurdle-rate rule [can] yieldapproximately correct decisions for these projects’’(p. 15). Focusing on an option-to-wait decision problem,and implementing variations in the cash-flow growth rate,the project volatility, and the discount rate, he shows that

for a wide range of project characteristics, fixed-hurdle rate and profitability index rule can providea good approximation to optimal investment timingdecisions, in the sense that the ex ante loss fromfollowing the suboptimal rule is small; it is possible tofollow the wrong investment rule without losingmuch of the ex ante value of the investment timingoption. In fact, as the investment timing optionbecomes worth more and it becomes optimal to waitlonger to invest, the option value becomes lesssensitive to errors in investment rules.’’ (p. 15)

He explains that even though the rise in the discountrate lowers the project value, it also lowers the value atwhich investment becomes optimal, so that ‘‘a decisionrule of the form ‘invest when the project has an internalrate of return of 20%’ might in fact be appropriate fora wide variety of projects’’ (p. 15). He underlines that ‘‘fora variety of parameters, particular hurdle-rate . . . rules canprovide close-to-optimal investment decisions. Thus, itmay be that firms using seemingly arbitrary ‘rules ofthumb’ are approximating optimal decisions’’ (p. 13,italics added). J&M (2002) reach the same conclusionholding that the use of hurdle rates should not be deemedless reliable than the use of the opportunity cost of capitalas a discount rate: ‘‘managers . . . [take] the right deci-sions . . . because they use a hurdle rate that is higher thanthe cost of capital to capture the option to wait’’ (p. 12).z

yAll following quotations from J&M (2002) refer to the Internet version.zSome other authors are puzzled by the ‘‘intriguing paradox’’ of such near-optimal choice behaviors: ‘‘we have compelling evidencethat managers . . . often do not use real option techniques . . .On the other hand, strategic decisions under uncertainty appear toconform to some general expectations based on real option theory . . .The resolution of this paradox would seem to be that, despitetheir biases, managers’ strategic investment decisions can loosely conform to normative real options models. Managers may employreal option reasoning, without getting all the details correct . . .Managers’ investment decisions may be ‘directionally correct’, even ifthey are not completely unbiased’’ (Miller and Shapira 2004, p. 281). However, the authors admit that actual choice behaviors takeaccount of variables disregarded by normative models: ‘‘normative models for pricing options overlook key aspects of thebehavioral and organizational contexts in which investment decisions occur’’ (p. 282).

Investment decisions, net present value and bounded rationality 969

Although these authors do not recommend the use ofaspiration levels and rules of thumb as decision making

criteriay it is worth noting that their conclusions implicitlyfoster the idea that advantages can be taken fromundertaking a simple heuristic:

[a] ‘‘the advantage of using a hurdle rate is thatmodelling all possible future options is not necessary’’

(J&M 2002, p. 4) and decision makers may ‘‘find ituseful to use a rule that best justifies makingintuitively plausible investment decisions’’

(McDonald 2000, p. 26).

[b] ‘‘managers adjust these rules . . .when an invest-

ment is strategic and expiring.’’ (McDonald 2000,p. 30).

[c] ‘‘use [of a hurdle-rate rule] in practice might stem

from the success of apparently arbitrary rules that arerevealed over time to be close to optimal. Managers

likely observe the capital budgeting practices, in theirown and other companies, and in many casesprobably mimic what seems to work’’ (ibidem).

It is worth noting that [a], [b] and [c] fit the threepremises on which Gigerenzer (2001) bases his notion of

adaptive toolbox:

[A] psychological plausibility: models of decision-

making should have adequate regard ‘‘for theconstraints in time, knowledge, and computationalcapacities that humans face’’ (Gigerenzer 2001, p. 38).

[B] domain specificity: ‘‘each heuristic is specializedfor certain classes of problems, which means that

most of them are not applicable in a given situation’’(Gigerenzer and Todd, 1999, p. 32) and ‘‘what worksto make quick and accurate inferences in one domain

may well not work in another. Thus, differentenvironments can have different . . . heuristics’’(Gigerenzer et al. 1999, p. 18).

[C] ecological rationality: this consists in ‘‘the matchbetween the structure of a heuristic and the structure

of an environment’’ (Gigerenzer and Selten 2001a,p. 9). Heuristics are ecologically rational in that ‘‘theyare adapted to particular environments . . . [and] can

be fast, frugal and accurate by exploiting thestructure of information in natural environments’’

(ibidem).

Sentences in [a] exemplify the psychological plausibility[A] of the NPV heuristic (decision makers are not able to

take into account all possible future opportunities andoutcomes). Sentence [b] reveals that these rules are

domain-specific [B] in the sense that when the contextchanges (strategic investments versus non-strategic

investments, expiring versus non-expiring projects) the

rule is changed or is combined with other rules of thumb(and that the aspiration levels are adjusted too, as

suggested by J&M. See below). For example, while the

NPV heuristic seems to work well for a variety ofindustrial projects, the recognition heuristic (Borges et al.

1999, 2008, Goldstein and Gigerenzer 1999, 2002) is

inapplicable in such frameworks. Also, the hurdle-raterule seems to work least well in the case of low volatility,

but the errors may be avoided through a combination of

rules (e.g. hurdle-rate and profitability index) to give riseto a different heuristic: ‘‘This suggests constructing a third

rule as a hybrid of the two rules . . .Such a hybrid rule can

prevent the large errors at extreme discount rates

generated by either rule alone . . . this example . . . demon-strates that firms in practice might find it useful

to . . . consider multiple rules at once’’ (McDonald 2000,

p. 26). Sentence [c] reveals that the use of the NPVheuristic is ecologically rational [C] due to success over

time. McDonald’s words also implicitly suggest that the

rules of thumb analysed may be the result of imitationand social learning. In other words, his words

encourage the view that the satisficing NPV rule is a

‘do-what-others-do’ (or ‘do-what-successful-people-do’)strategy (see Laland 2001; see also Berg and Lien 2003).

J&M (2002) cite empirical researches showing that

managers do not use the CAPM-derived cost of capital,

contrary to the normative suggestion of most corporatefinance textbooks. Decision makers apply hurdle rates

that are usually higher than the cost of capital. The value

of waiting to invest is underlined to explain this finding:

‘‘using high hurdle rates, companies in fact indirectlyaccount for the existence of timing options’’ (p. 15).

Resting on Poterba and Summers (1995) data, they show

that ‘‘a relatively high hurdle rate of 12.2% is successful incapturing most of the option value as long as the

uncertainty of projects’ cash flows is high’’ (J&M

2002, p. 18) and ‘‘If we take into account the option todelay the project, the financial decision is no longercrucially dependent on an exact figure of the discount

rate’’ (p. 19). However, there are other explanations,which suggest that the use of hurdle rates is consistent

with established theories in business and finance. For

example, a project may absorb managerial skills and thus

prevents firm from undertaking other profitable projectsin the future. Thus, ‘‘If managerial time of a skilful

manager is limited, she must decide when it is optimal to

take a project. It may pay off to wait and not take thenext best positive net present value project’’ (p. 21), which

explains why ‘‘the use of a hurdle rate that is higher than

the cost of capital . . . is likely to lead to near optimaldecisions’’ (ibidem). This also explains the large variation

that Poterba and Summers (1995) find in the hurdle rate

of different companies: firms in the same sector face

similar systematic risks and, therefore, if they compliedwith the CAPM-derived discount rate (which is based on

systematic risk), they would apply similar discount rates.

y‘‘We do not suggest that managers should use these rules of thumb’’ (McDonald 2000, p. 30).

970 C. A. Magni

When managerial resources are in limited supply it isappropriate to use higher discount rates. This view isconsistent with the resource-based theory and theliterature on Top Management Teams, according towhich managerial and firm-specific skills play a pivotalrole in value creation (Barney 1986, 1991, 2001, Grant1991, Grant and Robert 1995, Levinthal 1995, Bromiley2005). Furthermore, if a project is strategic, it may beworth undertaking it even if its NPV (computed with the‘correct’ cost of capital) is negative, because it promisesfuture opportunities. In this case the use of a discount ratesmaller than the cost of capital should be appropriate.Actually, J&M (2002) do affirm that ‘‘companies use lowhurdle rates for strategic projects’’ (p. 22); ‘‘Because it isdifficult to estimate the values of all possible upcominginvestment opportunities and to assign probabilities howlikely these will arise, companies seem to use a low hurdlerate that takes into account that the payoffs in the futureare possibly higher than the strategic investment itselfsuggests’’ (ibidem); ‘‘there are strategic situations wheremaking the first move has a commitment value’’ (Dixit1992, p. 119). The view is also consistent with the strategicmanagement literature, according to which strategicdecisions are drivers for competitive advantage (Collisand Montgomery 1995, Quinn and Mintzberg 1996). Forthese complex decisions the use of modelling is hardlyhelpful: ‘‘Even if highly simplified and abstracted, theassociated SA [strategic assets] decision may not besolvable in closed-form equilibrium . . .For example,when modelled as a differential game, the problem willprobably be not tractable’’ (Amit and Schoemaker 1993,p. 40). Agency theory (Jensen and Meckling 1976, Jensen1986) and the costs of external financing may also play arole in determining a hurdle rate different from normativecost of capital. J&M construct a comprehensive NPVfunction taking account of agency costs and the costs ofexternal financing. The authors claim that this is,according to some recent literature, the right NPVmodel to be maximized (pp. 23–27). That the costs ofexternal financing may play a major role is also suggestedby some empirical research about large reserves of cashretained in the firm (see Mikkelson and Partch 2003).

Brigham (1975) reports that 39% of respondentschange the hurdle rate less than once in a year (p. 20,Exhibit 4), and 32% state that it ‘‘depends on condi-tions’’: they ‘‘revise rates to reflect product and capitalmarket condition, with revisions generally occurring lessthan once a year’’ (p. 20). Gitman and Mercurio (1982)report that 50.3% of companies revise discount rateswhen environmental conditions change, 13% less fre-quently than annually, and 11.2% when a major project isevaluated (p. 27). The ‘product and market condition’seems to hint at Porter’s (1980) analysis, where variablessuch as rivalry, supplier power, buyer power, threat ofsubstitutes, and entry barriers are individuated as funda-mental drivers for value creation. As seen above, decision

makers raise the hurdle rate to account for the value ofwaiting, whereas strategic considerations tend to decreaseit; changes in project’s risk may well imply changes in thethreshold, but the notion of risk is not necessarilyequivalent to a normative one: ‘‘Threshold levels will berevalued . . . In this revising of threshold levels . . . projectswhich would have a minor loss if they failed would facea less stringent screening than those that could bankruptthe organization’’ (Carter 1971, p. 426). That is, non-ruinous losses versus ruinous losses are given relevance.This concern about risk is ‘‘consistent with the bounded-rationality approach of Simon (1957)’’ (ibidem) but hasnothing to do with usual capital budgeting textbookprescriptions: ‘‘the available models do not capture theessence of risk as defined by decision makers’’(Laughhunn et al. 1980, p. 1248).

Overall, these findings evidence that hurdle rates areseen as reference levels that do not change quickly but‘‘that may be gradually adjusted if they become too lax orbinding a constraint’’ (Goodie et al. 1999, p. 351).Historical documents also reveal that the hurdle-raterule, frequently used in past centuries, was used as a baselevel that could be adjusted: ‘‘If 15% tended to be the ruleof thumb, it was not applied slavishly by the viewers’’(Brackenborough et al. 2001, p. 144), who used to changeit ‘‘depending on the circumstances’’ (p. 143) (see alsosection 6 below). In other words, decision makerscondense into a base aspiration level several considera-tions related to uncertainty, decision flexibility, marketconditions, future opportunities, limited managerial skills,agency costs, and strategic considerations, but theheuristic is exploited in a non-rigid way so thatthe hurdle rate may change as the above factors change.

4. Problems in the unboundedly rational NPV

The unboundedly rational NPV rule has its own short-comings. First of all, there is no one NPV, buta proliferation of NPVs differing in terms of risk measureimplied, equilibrium model employed, and mathematicaltechnique adopted.y There is no agreement amongscholars on which one is the correct NPV. For example,‘‘academics do not agree on an appropriate equilibriummodel even for estimating firm-level discount rates, forwhich stock returns are observable; estimation of project-level discount rates is even more problematic’’(McDonald 2000, p. 21). Whatever the notion of risk,the equilibrium model, and the mathematical technique,many of the assumptions required to implement it areoften violated in real life. For example, the use of CAPM-derived cost of capital is often recommended in corporatefinance. Resting on a considerable amount of empiricalresults, J&M (2002) claim that ‘‘the CAPM, like allmodels, is only an abstraction from reality’’ (p. 5) and‘‘there have been many academic challenges to the validity

yThe fuzzy-logic version of the NPV is a further recent addition to the numerous existing versions (Ward 1989, Chiu and Park 1994,Abdel-Kader et al. 1998, Buckley et al. 2002). Admittedly, the cognitive implications of a cost of capital and a NPV based on fuzzylogic are an open issue.

Investment decisions, net present value and bounded rationality 971

of the CAPM as applied in practice’’ (ibidem). Lander andPinches (1998) find numerous disadvantages even in theuse of the more sophisticated real-options approach(expanded NPV). They write:

‘‘Any decision-making framework model is subjectto: inappropriate assumptions . . . poor estimationprocedures . . . failure to incorporate the effects ofcompetition and the strengths and weakness of thefirm into all aspects of the investment opportunity,ineffective information gathering of performance andmeasurement systems, inappropriate emphasis onshort-run goals and results, or excessive conservatismor optimism . . .Many of the required modelingassumptions are often and consistently violated ina practical real options application. The necessaryadditional assumptions required for mathematicaltractability limit the scope of applicability.’’ (pp. 542–543)

Furthermore, ‘‘The proper treatment of risk is stilla major source of concern’’ (Pinches 1982, p. 9) and theplethora of risk measures such as ‘‘standard deviation,beta, certainty equivalent, risk-adjusted discount rates,conditional probabilities, etc., can be illustrated but notcompletely agreed upon either in theory or in practice’’(Pinches 1982, p. 11). Among several others, a moststringent assumption frequently used is the completenessof the market. This property makes it possible toconstruct a replicating portfolio that mimics the project’spayoffs. In this case the expected rate of return of theportfolio is taken as the cost of capital and a positiveNPV indicates that, regardless of preferences, the investorbenefits from undertaking the project and selling short thereplicating portfolio. The approach is often called theoptions pricing approach. Barring the fact that short salesare often not allowed, arbitrageurs are nonethelessinevitably exposed to risk of losses (Shleifer and Vishny1997), contrary to what textbook arbitrage suggests.Above all, ‘‘without spanning, there is no theory fordetermining the ‘correct’ value for the discount rate �’’(Dixit and Pindyck 1994, p. 152). Smith and Nau (1995)admit that ‘‘without a market equivalent for the efficiencyuncertainty, we cannot construct a perfect replicatingtrading strategy or identify a unique risk-neutral prob-ability distribution and thus we cannot determinea unique option-pricing value for the project.’’ (p. 804).yThey illustrate an example where three strategies arepossible and conclude that ‘‘the optimal strategy isunclear and all three strategies are potentially optimal’’(p. 805). To put it in eco-logical terms: the assumptions ofthe normative NPV model do not fit the environmentwhere decision makers operate. Nor is its success in theenvironment ever tested: ‘‘Finance scholars teach net

present value is the right way to evaluate capitalinvestments. We have no research showing firms thatuse this technique do better than other firms . . .Theybelieve their theory so much that they do not need to testit’’ (Bromiley 2005, p. 70).

The heuristics-and-biases approach is a double-edgeweapon. For example, among the various equilibriummodels, ‘‘the predominant approach to estimating thecost of capital is to use the CAPM’’ (J&M 2002, p. 28).The CAPM is massively endorsed in corporate finance(e.g., Copeland and Weston 1988, Brealey and Myers2000, Koller et al. 2005, Damodaran 2006). ‘‘The CAPMprovides a framework for estimating the appropriateopportunity cost to be used in evaluating an investment’’(Rao 1992, p. 33). In a classical paper, Rubinstein (1973,footnote 10) proves that, if the CAPM assumptions aremet, the project is profitable if and only if

X1

X0� 14 rf þ

lcovð ~X1, ~rmÞ

X0,

with rf the risk-free rate, l the market price of risk, ~rm themarket rate of return, and cov the covariance.z Theauthor underlines that the cost of capital is the ‘‘appro-priate discount rate for the project’’ (p. 174) and that‘‘present value risk-adjusted discount rate . . . forms of thiscriterion are easily derived’’ (footnote 8, p. 171) as isshown by a simple manipulation of the above equation:

�X0 þX1

1þ rf þ lcovð ~X1; ~rmÞ=X0

4 0:

This decision rule boils down ‘‘to accepting the projectwith the highest net present value’’ (footnote 14, p. 174).It is worth noting that such a NPV is a disequilibrium(cost-based) NPV. Its use is recommended in popularfinance textbooks (e.g., Copeland and Weston 1983, 1988,Weston and Copeland 1988, Bøssaerts and Odegaard2001) and it is so widespread that Ang and Lewellen(1982) consider it ‘‘the standard discounting approach’’(p. 9) in finance. Magni (2007a) shows that the proceduresuggested by Rubinstein violates an accepted standard ofrationality: value additivity (warnings against the use ofthis NPV may also be found in Ang and Lewellen (1982)Grinblatt and Titman (1998) and Ekern (2006)).In particular, the author shows that the NPV ofa portfolio of projects is different from the sum of theNPVs of the projects (Magni 2007a, proposition 4.1). Thisimplies that the disequilibrium NPV provides differentevaluations (and different decisions) depending on theway the course of action is depicted. Loosely speaking, toreceive a 200E banknote or to receive two 100E bank-notes is financially equivalent: to employ the disequili-brium NPV means to be trapped in a sort of mentalaccounting (Thaler 1985, 1999) because evaluations differdepending on whether outcomes are seen as aggregate or

y‘‘We also have some semantic problems defining exactly what is meant by the value of a non-traded project’’ (Smith and Nau 1995,p. 804, footnote 7).zRubinstein’s proposal is logically equivalent to other classical proposals presented in the late sixties and early seventies by suchscholars as Mossin, Hamada, Tuttle and Litzenberger, Bierman and Hass (see Senbet and Thompson 1978 and Magni 2007b fora review).

972 C. A. Magni

disaggregate quantities. This amounts to saying that thedisequilibrium NPV is inconsistent with the principle ofdescription invariance, which prescribes that valuationsand decisions must be invariant under changes indescription of the same alternative. Violations of thisprinciple are known in the heuristics-and-biases traditionas framing effects (Tversky and Kahneman 1981,Kahneman and Tversky 1984, Soman 2004).y As opposedto the disequilibrium NPV, some scholars advocate theuse of an equilibrium NPV (e.g., Bogue and Roll 1974,Haley and Schall 1979), which is logically deducted fromthe CAPM as well. It is often expressed in a certainty-equivalent form:

�X0 þX1 � lcovð ~X1, ~rmÞ

1þ rf4 0:

This equilibrium NPV is additive, but Dybvig andIngersoll (1982) and Magni (2007a, c) show that, undercertain (realistic) assumptions on the market rate ofreturn, this NPV violates the no-arbitrage principle.z Thismeans that a decision maker complying with theequilibrium NPV is subject to arbitrage losses.Therefore, the equilibrium NPV is not consistent witha tenet that is considered a normative benchmark foreconomic rationality (Nau and McCardle 1991, Nau1999). As a result, the two CAPM-based NPVs displayinconsistencies with accepted standards of rationality.Rigorously speaking, one should say that the CAPM-based NPVs inevitably lead decision makers to infringeaccepted standards of rationality; but attention shouldnot be drawn to decision makers: decision makers whosebehaviors comply with this model do violate rationalbenchmarks, but it is not a fault intrinsic in the decisionmakers’ minds, it is a fallacy intrinsic in the very NPVmodels.

Given that no agreed-upon model exists in theliterature, ‘‘no general analytic solution to the full-blown financing and investment problem of the firm iscurrently available. The only recommendation that can bemade at present is that management must evaluate alloptions and do the best it can’’ (Pinches 1982, p. 12, italicsadded).

5. Normative role of bounded rationality

The use of hurdle rates in the NPV procedure bears strictanalogies to the tracking-error strategies of non-maximizer decision-makers, who compare their payoffto various ‘non-rational’ benchmarks. Focusing ona corporate context, Fernandez (2002) cites four suchbenchmarks for shareholders’ return: (i) zero (i.e. wealthincrease with respect to the beginning of the year),

(ii) return on treasury bonds, (iii) shareholders’ return ofcompanies in the same industry, and (iv) return of thestock market index (pp. 19 and 20). Others may be used,such as the investors’ own last return or average historicalreturn. Berg and Lien (2003), in an expected-utilityframework, show that tracking-error decision-makerswho, in addition to mean and variance, care abouta subjectively determined benchmark, obtain greatershares of accumulated wealth than pure mean-variancedecision-makers. In analogy to this result, in value-basedmanagement Young and O’Byrne (2001) explicitly suggestthat managers’ compensation plans should be based ona historical benchmark such as an increase of residualincome rather the level of residual income, because anincentive scheme based on excess-profit improvement ismore appropriate to align managers’ interests to share-holders’ interests.x

Cognitive illusions and cognitive limits themselves maybe beneficial: Berg and Lien (2005) show that anoverconfidence bias of non-expert decision-makers mayshift a standard financial market equilibrium intoa Pareto-superior equilibrium, because over-tradingleads to improved liquidity and lower transaction costs.Similar results are found by De Long et al. (1991), Kyleand Wang (1997), Dekel (1999), Gintis (2000) andHirshleifer and Luo (2001). Borges et al. (1999) showthat ignorance-based investment decisions grounded onthe recognition heuristic may outperform funds managedby experts. Berg and Gigerenzer (2007) show that it is justa satisficing behavior of the same kind we investigate inthis paper that leads regulators to intervene less than theywould have to in a world in which all agents weremaximizing neoclassical objective functions. It remains anopen issue to ascertain which violations of standardrationality are beneficial and in which environment: this isjust the normative aim of ecological rationality, whoseliterature is collecting more and more evidence on howheuristics may engender better decisions (Todd andGigerenzer 2007). This evidence ought to induce scholarsto rethink the norms (Gigerenzer 2006), that is, to accepta normative role of behavioral economics and, more ingeneral, of psychology. While a descriptive falsification ofunbounded rationality models is the accepted role ofpsychology among the communities of economists andpsychologists, a positive role for normative justification(corroboration/falsification) of decision-making modelsand theories may be fruitfully fostered. An explicit stanceon this issue is taken by Berg (2003), who endorses the useof behavioral economics as a normative tool. As regardsthe NPV model, in particular, we have seen that not onlydoes the maximizing model lead to violations of acceptedstandards of rationality (see previous section), but alsothat a boundedly rational perspective (satisficing) mayprove successful. And it is also worth noting that scholars

ySee also Haley and Schall (1979, pp. 182 and 183) for the unreliability of the disequilibrium NPV in ranking projects.zMagni (2007c) focuses on CAPM-based capital budgeting and shows that the derivative of the NPV function may be decreasingwith respect to the end-of-period cash flow in some state of nature. Dybvig and Ingersoll (1982) focus on asset pricing and show thatCAPM does not guarantee the absence of arbitrage in a security market.xSee Magni (2009) for an analysis of the notion of residual income.

Investment decisions, net present value and bounded rationality 973

themselves are inclined to support the expanded NPV onthe basis of its capability of explaining deviations ofactual behaviors from the normative (traditional) NPV(see section 6).

Cosmides and Tooby (1994) complain that ‘‘psychol-ogists are called in only to provide second-order correc-tions to economic theory’’ (p. 327). The growing amountof evidence in the literature of recent times shows thatpsychology and behavioral economics may be called in toprovide first-order corrections as well, to the benefit ofeconomics and, more importantly, of actual decisionmakers.

6. NPV rule: heuristic or unboundedly rational model?

The classical idea that cognitive processes may be dividedinto two main classes (Gilovich et al. 2002) is of someconcern in cognitive psychology. According to this view,mental processes make use of a heuristic-based system ofreasoning and of a rule-based system of reasoning, theformer being associative and intuitive, the latter beinganalytic and reflective. The former employs automatic,effortless, rapid strategies for decision-making, the latterrests on controlled, effortful, deductive processing (Evans2004, Sloman 1996a, b, 2002, Kahneman and Frederick2002). Each of the two systems has its specific domain,but it is plausible that the two systems are interactive and‘‘function as two experts who are working cooperativelyto compute sensible answers.’’ (Sloman 2002, p. 383).However, one never knows where either system beginsand where it ends, and the distinction between the two israther fuzzy: it is not easy to find clear-cut differencesbetween them (Gigerenzer and Regier 1996).

This is actually testified by the NPV model we arecoping with. The NPV model is presented in the literatureas the solution of an optimization problem (e.g., Fisher1930, Rubinstein 1973, Dixit and Pindyck 1994, McMinn2005). Still, there is evidence that it is a mixed methodol-ogy, a blending of logical and ecological rationality. Asmay be inferred from several historical surveys, it is theresult of rigorous mathematical thinking associated witha more intuitive and adaptive thinking of skillful decisionmakers. In a period when the quest for axiomatic modelswas cogent, Fisher (1930) just gave an optimization dressto a technique that was practiced long since: sinceFibonacci’s present-value analysis in the 12th century(Goetzmann 2005), contributions from mathematicians,philosophers, engineering economists, and actuarialscientists nourished and were nourished by the non-optimizing evaluations of properties, trees, lands, colli-eries, coppices, buildings, leases, shops, etc. made by thelegal, banking, and business communities (Wing 1965,Edwards and Warman 1981, Miller and Napier 1993,Scorgie 1996, Brackenborough et al. 2001). The use ofa systematic discounting procedure adjusted to reflectpersonal judgment is exemplified in actual evaluations

accomplished across centuries: Scorgie (1996) shows thatsince the 13th century ‘‘the capitalised value or selling

price of arable land and other property was determined by

multiplying the annual rental by some generally acceptedfactor’’ (p. 240, italics added), which is interpretable as

a discount factor. A valuation of Lord Cromwell’s

property in 1469 was obtained by using a factor of 20.yThe latter only ‘‘provided a base valuation’’ (ibidem) and

could be changed depending on the case at hand: ‘‘Theprovision for the use of another rate provided some

leeway’’ (Scorgie 1996, p. 240). Variations were governed

by several domain-specific factors, among which ‘‘theneed to use a higher interest rate where an investment

project involved hazards and uncertainty’’ (p. 242). For

example, the risk of flooding for a coal mine (ruinousevent) could suggest decreasing the factor from 20 to 4 or

to 3 (i.e. raise the discount rate to 25% or 33.33%)(p. 244). Brackenborough et al. (2001), focusing on the

Tyneside coal industry in the 18th century, report the use

of a NPV rule where cash flows ‘‘were discounted atinterest rates to reflect the viewer’s assessment of risk’’

(p. 141), which means that the ‘‘responsibility for

assessing the degree of risk was placed firmly in thehand of the client’’ (p. 144). Quoting directly from

original records, the authors confirm that the discountrates reflected ‘‘the purchasers’ own ideas of the risk and

prospects of the concern’’ and that the decision maker

should ‘‘judge for himself, and draw his own conclusions’’(p. 144). Analysts just provided the formal framework,

while leaving investors free to use the rule subjectively

‘‘through adjustment of the discount rate’’ (pp. 150 and151). The use of a rigorous procedure combined with

personal evaluation is also confirmed by Wing (1965),who analyses a paper by Van Deventer (1915) where

a foreman provides the owner of a small machine shop

with what can be seen as an unstated NPV criterion:‘‘First, estimate the probable saving that the investment

will make. . .Second, assign a probable length of life to

it . . .Third, estimate what the investment willcost . . .Fourth, pick out a minimum rate of return that

you will expect’’ (Wing 1965, p. 475). That minimum rateof return is a ‘‘desired profit rate’’ (ibidem) in a satisficing

sense. That the choice of the rate was meant to be

subjective is confirmed by the table Van Deventerprovides in his paper: it ‘‘was supplied with a range of

rates of return into which he [the shop owner] could fit his

personal minimum expected rate’’ (Wing 1965, p. 479,italics added). This cooperation is brought into play even

in the immaculate reign of unbounded rationality, thoughhardly recognized: when suggesting the use of dynamic

programming, Dixit and Pindyck (1994) explicitly leave

decision makers free to set their own desired discountrates: ‘‘dynamic programming can . . . be used to max-

imize the present value of the firm’s expected flow of

profits, subject to an arbitrary discount rate’’ (p. 149,italics added); ‘‘we can . . . use a dynamic programming

approach with an exogenously specified discount rate’’

yThe implicit discount rate is 5%¼ (1/20) � 100.

974 C. A. Magni

(p. 185); ‘‘The dynamic programming approach started byspecifying the discount rate, �, exogenously as a part ofthe objective function’’ (p. 120). To presenta maximization problem where the objective functiondepends on an arbitrary (thus subjective) hurdle ratemeans to suggest a blending of an unboundedly rationalprocedure and a boundedly rational heuristic.Interestingly, the authors themselves are puzzled bytheir own proposal: ‘‘One problem with this investmentrule is that it is based on an arbitrary . . . discount rate �.It is not clear where this discount rate should come from’’(p. 147). Yet, their classical book heavily (and commend-ably) relies on this hybrid NPV rule. Even McDonald(2000), in presenting the maximization procedure, sig-nificantly admits: ‘‘We will be agnostic about thedetermination of �’’ (p. 16).

One may even conjecture that the birth of thesophisticated version of the NPV rule (the expandedNPV) has been favored by empirical research: the lattermay have induced scholars to develop a new enhancedNPV capable of explaining the empirical deviations fromthe traditional NPV rule. This is drawn from their explicitconcern about an explanation of actual choice behaviors:

Some recent developments in the theory of invest-ment under uncertainty have offered an interestingnew explanation of these phenomena . . .This newapproach suggests that textbooks pictures of thedynamics of a competitive industry need . . . substan-tial redrawing’’. (Dixit 1992, p. 108, italics added)

The real options literature can explain why usinga discount rate that is higher than the cost of capital isa good strategy when waiting opens up betteropportunities. Such a discount rate is typicallyreferred to as the hurdle rate. (J&M 2002, p. 13,italics added)

The option insight also helps explain why the actualinvestment behaviour of firms differs from thereceived wisdom taught in business schools. Firmsinvest in projects that are expected to yield a return inexcess of a required, ‘‘hurdle’’ rate . . . such hurdlerates are typically three or four times the cost ofcapital . . . firms do not invest until price risessubstantially above long-run average cost. On thedownside, firms stay in business for lengthy periodswhile absorbing losses, and price can fall substantiallybelow average variable cost without disinvestment orexit. This also seems to conflict with standard theory,but . . . it can be explained once irreversibility andoption value are accounted for. (Dixit and Pindyck1994, p. 7, italics added)

We found qualitative implications, and severalapproximate quantitative ones, that conform toexperience—the high hurdle rates used by businessfirms when they judge investment projects, therelative inefficacy of interest rate cuts as policies tostimulate investment projects, the significant and

detrimental effect of policy uncertainty on invest-ment, and so on. (Dixit and Pindyck 1994, p. 425)

. . . there is considerable anecdotal evidence that firmsmake investment decisions in a way that is at leastroughly consistent with the theory developed in thisbook (for example, the use of hurdle rates that aremuch larger than the opportunity cost of capital aspredicted by the CAPM). (Dixit and Pindyck 1994,p. 296, italics added)

These claims may be interpreted as a regard forproviding a theory that should conform to decisionmakers’ actual choices: ‘‘note that options advocatesjustify options models over [traditional] net present value(NPV) models precisely because the assumptions of theoptions models fit the reality better than the assumptionsof the [traditional] NPV model’’ (Bromiley 2005, p. 69,italics added). This means that scholars themselvesimplicitly (and unawarely) endorse a normative role forpsychology and behavioral economics. Thus, if decisionmakers do not abide by a normative theory, they shouldnot be tagged as irrational. Rather, one could infer thattheory exhibits ‘‘failures’’, ‘‘anomalies’’ (Dixit andPindyck p. 4), ‘‘abstraction from reality (J&M, p. 5),and ‘‘misleading results’’ (Bromiley 2005, p. 69). Decisionmakers’ heuristics may give insights for the constructionof new theories and models that take account ofinformation, context, environment and personal evalua-tion. Unbounded rationality alone is not sufficient:

Any decision-making framework, however, can onlyimprove a manager’s understanding of the problem athand and help him/her to make a more informed andconsistent decision. No decision-making frameworkcan guarantee a ‘‘good’’ outcome and there is nosubstitute for managerial effort, creativity, experience,knowledge, and critical thinking . . .’’ (Lander andPinches 1998, p. 542, italics added).

The ‘‘environmental indifference that is often assumedby many financial theories and models is not a reasonableassumption for the future’’ (Crum and Derkinderen 1981,p. 237, italics added): the quest for a cooperation ofbounded and unbounded rationality is compelling.

7. Concluding remarks

The heuristics-and-biases program draws attention tological fallacies of human reasoning that cloud humanminds (Tversky and Kahneman 1974, Kahneman et al.1982, Kahneman and Tversky 1996): choice behaviors arebiased because they deviate from accepted standards ofrationality. By contrast, advocates of bounded rationalityare concerned with showing that heuristics may oftenprove useful in decision-making (Gigerenzer et al. 1999).The latter aver that cognitive illusions disappear onceheuristics are seen as adaptive tools rather than logicaldevices for solving decision-making processes (Gigerenzer

Investment decisions, net present value and bounded rationality 975

1996, 2000). Gigerenzer and the ABC group endorse theidea that human beings’ rationality is ecological ratherthan logical, and their fast-and-frugal-heuristics programaims at presenting a number of heuristics that aresuccessfully applied in real-life in specific environments.This work deals with the NPV maximization model,which is a keystone in economics and finance. The mainresults may be summarized as follows:

(1) real-life decision makers often use the NPVcriterion, but discount cash flows with a hurdlerate that differs from the cost of capital norma-tively suggested (e.g. CAPM, arbitrage pricing,multifactor models);

(2) the hurdle-rate rule may be interpreted asa boundedly rational approach to investmentdecisions: decision makers rest on aspirationlevels subjectively determined;

(3) some empirical evidence shows that the bounded-rationality approach is ecologically rational,domain-specific, and psychologically plausible.The hurdle-rate rule leads to close-to-optimalsolutions when confronted with the expandedNPV;

(4) several factors affect the hurdle rate, among whichare decision flexibility, future opportunities, ration-ing of managerial skills, strategic considerations,agency costs, and costs of external financing. Riskis another factor, but the notion of risk is notequivalent to the risk measures employed inanalytical models;

(5) consistency with several theories and models ishighlighted, among which real options approach,resource-based theory, Top Management Teamliterature, agency theory, and strategic manage-ment literature. The hurdle rate is set at a base levelbut is not rigidly applied: possible fluctuationsaround the base level depend on various domain-specific and project-specific factors (such as thedrivers just mentioned);

(6) the NPV maximizing model is not univocal: it maytake on several different dressings. The use ofa particular version depends on the circumstances:for example, the arbitrage-pricing NPV may not beused if the actual market is not complete. Some ofthe versions are often inapplicable and/or revealinconsistencies with accepted standards of ration-ality (additivity, no-arbitrage, descriptioninvariance);

(7) a normative role for psychology and behavioraleconomics may prove useful in the construction ofnew theories and revision of older ones on the basisof actual behaviors;

(8) an interaction between the two systems of ration-ality may improve performance. Actually, the verydistinction between the two systems of reasoning israther artificial and only useful as a metaphor (seeGigerenzer and Regier 1996): as attested by recenthistorical research, the origins of the NPV just liein the common efforts of various categories of

people facing the need of both logical and eco-logical rationality.

This paper may suggest a direction for research andhopefully will act as a stimulus for further inquiries. Thescientific niche that might be disclosed could offerunexpected views about bounded and unbounded ration-ality and their interrelations. A possible payoff of sucha view is that bounded rationality and unboundedrationality are not necessarily rivals. After all, unboundedrationality itself, as derived from logic and mathematics,may not abstain to consider itself as a derivation ofecological rationality: logic and mathematics should beintended as the most advanced step of human simulation,and simulation is a tool for anticipating; as such, it isindispensable for discovering and creating. The impres-sive degree we have achieved in such an ability may beseen as the result of an evolutionary process, in which itssurviving value has been adaptively tested. Therefore,logic and mathematics, as symbolic tools, assemble theexperience of our ancestors (Monod 1970, pp. 171 and172). To put it in a nutshell, even logic is ecological.

More ‘fluid theories’ and more mixed methodologieswill help us understand decision makers and help decisionmakers cope with complex decision problems, with thepleasant by-product of conciliating the two rival parties.

Acknowledgements

The author wishes to thank Nathan Berg for hisinvaluable remarks and suggestions.

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