8. Ranking Errors in CAPM Capital Budgeting Applications

7/29/2019 8. Ranking Errors in CAPM Capital Budgeting Applications

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Ranking Errors in CAPM Capital Budgeting Applications

Author(s): Richard J. Rendleman, Jr.Reviewed work(s):Source: Financial Management, Vol. 7, No. 4 (Winter, 1978), pp. 40-44Published by: Wiley on behalf of the Financial Management Association InternationalStable URL: http://www.jstor.org/stable/3665084 .

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Ranking E r r o r s in C A P MCapital Budgeting ApplicationsRichard J. Rendleman, Jr.

Richard . Rendleman, r., is AssistantProfessorof Financeat the GraduateSchool of Managementat NorthwesternUniversity.

* The Sharpe-Lintner-MossinapitalAsset PricingModel (CAPM) providesa framework or valuingcertaintypes of financialassets in a perfectand ef-ficientcapitalmarket.Rubinstein6],Weston 8],andothershaveapplied he modelto the firm'scapitalex-pansiondecision o determineheequilibriumalueofthe financialclaimsthat are used to finance a par-ticular asset under consideration.If this value isgreater hanthe asset'spurchase rice, henthewealthpositionof the firm's current hareholdershould n-creasebythe differencenthe two values.Ontheotherhand, if the capitalizedvalue of the project'scashflows is less than its cost, the projectshould be re-jected,and the firmshouldconsideralternativemeansof expansion.The use of the CAPM for capitalbudgetinghasrecently omeunderattackby MyersandTurnbull5]and Fama [1]. These authorshave shown that theCAPM cannot alwaysbe effectivelyemployedas amethod ordeterminingheproperdiscountrate foraproject. For example, Fama shows that in orderproperly o employthe CAPM to determinea con-

stantrequired ate of return, here can be no uncer-taintysurroundinghe future iskless nterestrate,themarketprice of risk, or the systematicrelationshipbetween proportionalchanges in the project'sex-pectedcash flows and the marketreturn.Expectedcash flows are the onlyparameterwhosevaluecanbeuncertainthrough time. Moreover, expected cashflows must evolvethrough ime as a martingale. notherwords, heexpected ashflowat anytimet mustbe the best estimate for the cash flow at t + 1.Myers and Turnbulldemonstrate hat, if a proj-ect's cash flows do not evolve as a martingale, heproject'sbeta willdependuponthe life of the projectand the growthrateof the cash flows.Moreover, heobserved ystematic iskof thefirmshouldreflect heriskof its tangibleassetsas wellas its more uncertainfuturegrowth opportunities.When all these factorsare considered ogether, t is unlikelythat observedbetas will be representativef the "true"systematicrelationship etween he firm'scash flowsand overalleconomicactivity.Thispaperassumes hat the conditions pelledout

o 1978 Financial Management Association 40

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RENDLEMAN/RANKING RRORS

by MyersandTurnbull ndFamafor properlyusingthe CAPM fordetermining project's equiredateofreturnare met. Evenunder heseconditions,potentialerrorscanarise whenrankingprojectson thebasisoftheir expectedexcess returnsif market-determinedmeasuresof systematicrisk rather than measuresreflectingprojectcost areemployed o determineherequiredrates of return.Thus, anotherchallenge spresented o the practicalusefulnessof the model.Project Selection

Rubinstein as shown hattheequilibriumalue ofa project,Vj, will be greater han its cost, Cj, if thefollowing nequalityholds:Xj- > + RF+ XCOV (C ,m), (1)cj jwhereXj is a randomvariablewith meanXj denotingtheend-of-periodalueof theproject,Am is thereturnonthemarketportfolio,RF s theriskless nterest ate,X s the"marketpriceof risk"which s assumedo beconstantacrossall financialassets,andCOV(., .) isthecovariance perator.Thiscriterion ndicateshat aprojectshould be accepted f its expectedreturn shigher hanthatof otherassets of equivalent iskthatsell forequilibrium rices.Inequality 1) canbe alter-nativelystated in a morefamiliarformas

> 1+RF+/ (E[r ]-RF)ciwhere is the betaof the asset'sreturnswithrespectto the returnon the marketportfolioand E (.) is theexpectationoperator.Since

COV ( ,j m)1V= j where VAR (Am) is the vari-VAR (Am)ance of the marketreturn, henE( ) - R F . Thus, all references to the sys-VAR (Rm)

tematic risk of a project,measured n terms of itscovariance, an be alternatively iewedas a referenceto the project'sbeta.Actually, hecovarianceerm n (1) does not repre-sent the equilibrium ystematicrisk of the project.Assuming that the project's equilibriumvalue isgreater han ts cost,thecovarianceermwilldecreasefrom COV ( j, Am) to COV Xj , fm) when theCj Vjproject s capitalizedat its equilibriumalue. Wewillrefer to the former term as the disequilibrium

covariance and the latter as the equilibriumcovariance.The problemaddressedn this analysisarisesfromthe difference betweenthese covariances.The dis-equilibrium ovariancemust be estimated on an exante basis, which is quitedifficultto do in practice,because t involvesthe estimationof the covariancebetween heproject's nd-of-periodash flow andthemarketreturn a numberwhichfew can graspin-tuitively.Even if the covarianceerm is transformedintothemorefamiliarbeta,the samecalculationmuststill be made.To avoid having to make this type of ex antecalculation,one mightlook to the securitiesmarketand measurecovariancesor betas from historicalreturndata of firms with economiccharacteristicssimilarto those of the projectunderconsideration.The problemwiththis approachs that measuresofsystematicrisk so obtainedare likely to be betterproxiesfor the equilibriummeasure(assuming hatthe market is in equilibrium)than for its dis-equilibrium ounterpart.Fortunately, o errorwillresult f oneinterchangesthe covariancesn (1) and as a basis for the accept-rejectdecisionusesa market-determinedequiredateof returnof assetssellingfor equilibrium riceswithsimilar economic characteristicsas the project. Aprojectwill have a positivenetpresentvalue f its ex-pectedreturns greater han ts requiredate of returnusingthe equilibriumovarianceas in (2), below.*

> + RF+ XCOV ( m)ci Vi, (2)However, f projectsare ranked on the basis of ex-pectedexcessreturns,and theequilibriumovarianceis used,potentialrankingerrorscan result.Project Rankings

In theabsenceof capitalrationing, firmshouldbeunconcernedwithprojectrankingsunless heprojectsunderconsideration remutually xclusive.The firmshould simply invest in those projectsthat have apositivenet presentvalue.*If a projectdoes not have a positivenet presentvalue, then Cj > Vjandx 5:< ( + RF)+ XCOV( , Am).Cj CjSince Vj < C, we can substituteVj forCj in the covariance term andstill maintain the inequality. As a result, criterion (2) cannot giverise to an incorrect accept-reject signal for a project.

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Whether or not we believe that capital rationingcanexist in an efficient capital market, it appears thatmany firms operate in a capital rationing environ-ment. For example, in a recent survey appearing inFinancial Management, Gitman and Forrester foundthat approximately50% of the firms surveyedoperateunder capital rationing and "attempt to allocate afixed budget on a competitive basis" [2, p. 69].Even in the absence of capital rationing, it wouldappear that firms ought to be concerned with projectrankings,because projectswith the highest net presentvalues are likely to be the least sensitive to exactspecifications of expected cash flows and costs ofcapital. Moreover, even if these estimates are made asprecisely as possible for a given economic environ-ment, the most profitableprojectsare least likely to berejected, given a change in the economic setting.Therefore, it is importantto understand he conditionsunder which a project selection criterion ranks proj-ects correctly.A formal analysis of the ranking problem ispresented in the appendix. The problem does not oc-cur when projects are evaluated using net presentvalues calculated at rates based on equilibrium ormarket-determined measures of systematic risk. Itdoes not occur if the excess of internal returnover re-quired return is used in ranking projects of the samescale when requiredreturn reflects the systematic riskassociated with project purchase price ratherthan themarket value of the projectin place. Ranking becomesa problem when requiredreturn is calculated conven-tionally by using an equilibriumor market-determinedmeasure of systematic risk and comparing this figureto internal return.Illustration

An illustration, involving two one-year projects ofthe same scale, i andj, should help to make the issuesclear. In the illustration, the risk-free rate, RF is .05,and the "market price of risk," X, is 1. The informa-tion on projects i and j is as follows:Project Project

ExpectedEndof PeriodValue,XCovariance f EndofPeriodValuewithMarketReturn,COV X, lm)Equilibrium rojectValue,V = [X- X COV (fX, m)]/(1 + R,)Costof Project,CNetPresentValue,NPV

$125 $150

$ 14.75 $ 42.90$105$ 8817

$102$ 8814

Project i has the highernet presentvalue and wouldbe the higher ranking of the two projects on this basisand the better mutually exclusive choice for a wealth-maximizing firm.However, if the firm were to rankthese two projectson the basis of excess of internal return overequilibrium or market-determined return, then thefirm would be misled:

Project ProjectCovariance f EquilibriumReturnwithMarketReturn,COV (X, tm)/VRequiredReturnBasedonEquilibriumrMarket-

Determined ystematicRisk,Rf + X[COV ,f, m)/V]InternalReturn,X/C) - 1Excessof InternalOverRequiredReturn

.1405 .4206

.1905 .4706.4204 .7045

.2299 .2339Ranking on this basis favors Project j. If the rank-ing were used to choose between these two projectsthen, clearly, wealth would not be maximized.The problem can be solved with an excess returnranking system, but the solution requires that the ac-quisition cost of the project be recognized incalculating required return:

Covariance f ActualProjectReturnwithMarketReturn,COV (Xt, m)/CRequiredReturnBasedonAcquisitionCostDeterminedSystematicRisk,Rf +X [COV(,, lm)/C]InternalReturn,X/C) - 1Excessof Internal verRequiredReturn

Projecti Project.1676 .4875

.2176 .5375.4204 .7045

.2028 .1670With required return based on project acquisition

cost, the ranking is correct.This problem exists for any capital asset pricingmodel that provides risk-adjusted project screeningrates. The implication for the Sharpe-Lintner-MossinModel is that one must estimate the risk of a projectas a function of the riskiness of its own cash flows andnot that of another similar asset selling for anequilibrium price. Therefore, it is inappropriatefor afirm to use its own beta when computing the expectedexcess return of a project with risk characteristicsidentical to the firm itself. In the same way, it is inap-propriateto use the beta of any firm involvedin an ac-tivity analogous to a project in computing excess

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return. This presents a problem for the practicalapplicability of the model because the required ac-quisition cost-based estimates of systematic risk aredifficult to make on a subjective basis. On the otherhand, if the concern in capital budgetingis whether ornot a project should be acceptedor rejected,a market-determined measure of systematic risk can beemployed to determine a project's required rate ofreturn.References1. Eugene F. Fama, "Risk Adjusted Discount Rates andCapital BudgetingUnder Uncertainty,"Journal ojFinancial Economics (August 1977), pp. 1-24.2. LawrenceJ. Gitman and John R. Forrester,Jr., "ASurveyof CapitalBudgetingTechniquesUsedbyMajorU.S. Firms," Financial Management (Fall 1977), pp.66-71.3. John Lintner,"The Valuationof Risk Assets in StockPortfoliosandCapital Budgets,"Reviewof Economicsand Statistics(February1965),pp. 13-37.4. Jan Mossin,"Equilibriumn a CapitalAsset Market,"EconometricaOctober1966),pp. 768-83.5. Stewart C. Myers and Stuart W. Turnbull,"CapitalBudgetingand the CapitalAsset PricingModel:GoodNews and BadNews,"Journalof Finance May 1977),pp. 321-32.6. MarkRubinstein, A MeanVarianceSynthesisof Cor-porateFinancialTheory,"Journalof Finance(March1973),pp. 167-81.7. WilliamF. Sharpe,"CapitalAsset Prices:A TheoryofMarketEquilibriumnderConditions f Risk,"Journalof Finance September1966),pp. 425-42.8. J. Fred Weston, "InvestmentDecisions Using theCapitalAsset PricingModel,"FinancialManagement(Spring1973),pp. 25-33.Appendix.

Considerwo projects, and , both of whichcan bepurchasedor thesameprice,C. Byemployinghe dis-countedcertaintyequivalentversion of the CAPM,project has a highernet presentvalue thanj, ifXi - COV (Xi, m) > Xj - COV (j, Afm)1+ RF 1 + RF

(A-l)By multiplying (A-l) by 1 + RF and dividing by thecost of theprojects,we findthatproject has a highernet presentvalue if

-i- X COV ( tm)> -X COV (- ,m).(A-2)

Finally, if 1 + RF is subtractedfrom both sides of (A-2), the inequality can be expressed in terms of ex-pected excess returns:X ( +XC)>- ([1 + RF]+ XCOV[ ,alm])>c cxj -( ([1+ RF]- XCOV [--,-IM). (A-3)Thus, project i has a higher net present value thanj ifits expectedexcessreturn s higher han that of j.Note that the covarianceerms of (A-3) are basedupontheanticipated eturns f theprojects alculatedwith referenceo theirpurchaseprices,whichmaybedifferent rom theirequilibrium rices.If a differenceexists, the covariance terms will not be the same asthose obtained by measuring the systematic risk ofsimilar assets that sell for equilibrium prices.If one ranks projects on the basis of their expectedexcess returns, it is possible that the rankings will beincorrect if market-determinedequiredrates ofreturnareemployed.To illustratehepotentialerror,considera situation n which both i andj have thesame cost, C, but project i has a higher equilibriumvalue than project j. In addition, assume that bothprojects have a positive net present value, in whichcase C < Vj < Vi. If market-determinedequilibriumexpected returns are used in the computations, anerror will result if

Xj _ Xj Xi XiC Vj C Vi, (A-4)InequalityA-4) states thatproject has the higherexpectedexcess return.However,we have assumedthat i has the highernet presentvalue.Therefore,apotentialrankingerrorcan occu0if it is possible orboth conditions to hold simultaneously. Under whatcircumstanceswill this be possible?By solving (A-4) in terms of C, projectj will have a

higherexpectedexcess return f(A-5)(j - Xi)X XiJ v1

providedthat the quantity (- -X) is positive. Oth--j Vierwise, the inequality will be reversed. If Vj < Vi, wecan only be assured that this quantity is positive if Xj> Xi. In addition, the fact that Vj < Vi implies that C< Vj < Vias longasXj > Xi. Thiscan be seenbysub-stituting Vj for Vi in (A-5) and cancelling (Xj - Xi).The inequality hen becomes C < Vj.

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If (V'- X') s negative, he inequalitywill be re-vj Viversed.Sinceit has beenassumed hatproject has ahigher equilibrium alue thanj, Xi must be greaterthan Xj to ensure that this quantity is negative.However,under heseconditions,he fact thatVj< Viimplies hat C > Vi > Vj,or thatneitherprojecthas apositivenetpresentvalue.Therefore,t is notpossiblefor rankingerrors o occur for projectswithpositivenet presentvalues f their costs are the same and theprojectwith thehigherexpected xcessreturnprojectj in thisanalysis)has the lowerend-of-period ayoff.Althoughit is possiblethat a rankingerrorcanoccur f theprojectwiththehigher xpected xcessre-turn has the higher expectedend-of-periodpayoff(i.e., Xj > Xi), this is not a sufficient condition to

cause an error. The final factor to be consideredis the relationshipbetween the covarianceterms,COV (Xj, ftm)and COV (Ri, km), with Xj > Xi. Sincethe equilibriumvalue of any project is given byX - COV (X, Rm) , the only way that Vj can be less1 + RFthanVi,andXjbegreater hanXi, is if COV(&j, m)> COV (Xi, Atm).To summarize,no rankingerrorcan occurif thedisequilibriumovariances used o calculate xpectedexcess returns.If the equilibriumovariances used,no errorcanoccur f eitherXj < Xi or COV(Xj, tm)< COV(Xi, tm), where is theprojectwiththehigherexcessreturn.However,f both of theseconditions resatisfiedalongwith(A-5),then hetwoprojectswill beranked ncorrectly.

FINANCIAL MANAGEMENT ASSOCIATIONNINTH ANNUAL MEETINGCALL FOR 1979 PROGRAMThe Financial ManagementAssociation brings together practicingfinancial managers fromindustry, inancial nstitutions,and nonprofitandgovernmental rganizations, nd membersof theacademiccommunitywith interests n financialand investmentdecision-making.The ninth annualprogram,October11-13, 1979,at theSheratonBostonHotelin Boston,Massachusetts,willstress heinterrelationshipsetween heoryandpracticen financialandinvestmentmanagement.Proposals oparticipatenthe formof completedpapersortwo-page maximum) bstracts re solicited orthe 1979

meeting.Studentcontributions reencouraged.All papersandabstracts houldbe receivedno laterthanFebruary 8, 1979.Bothmembers ndnon-membersre nvited o respondo the Vice-President-Program,GeorgeH. Hempel,Cox Schoolof Business,SouthernMethodistUniversity,Dallas,Texas75275, Telephone (214) 692-2590.1979 Annual Meetings

Dates: October 11-13, 1979Place: SheratonBoston Hotel

Boston, MassachusettsProgramParticipation:

MeetingArrangements:

Placement nformation

ProfessorGeorgeH. HempelEdwinL. CoxSchoolof BusinessSouthernMethodistUniversityDallas,Texas75275Tel:(214)692-2590ProfessorFrankCampanellaExecutiveVice-PresidentBostonCollegeChestnutHill,Massachusetts2167Tel:(617)969-0100ProfessorDonaldJ. PuglisiCollegeof Business&EconomicsUniversity f DelawareNewark,Delaware19711Tel:(302)738-2556

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8. Ranking Errors in CAPM Capital Budgeting Applications