Admati Et Al 86 on Timing and Selectivity

8/12/2019 Admati Et Al 86 on Timing and Selectivity

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American Finance Association

On Timing and SelectivityAuthor(s): Anat R. Admati, Sudipto Bhattacharya, Paul Pfleiderer and Stephen A. RossReviewed work(s):Source: The Journal of Finance, Vol. 41, No. 3, Papers and Proceedings of the Forty-Fourth

Annual Meeting of the America Finance Association, New York, New York, December 28-30,1985 (Jul., 1986), pp. 715-730Published by: Wileyfor the American Finance AssociationStable URL: http://www.jstor.org/stable/2328504.

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THE JOURNALOF FINANCE* VOL. XLI, NO. 3 * JULY 1986

On Timing and SelectivityANAT R. ADMATI, SUDIPTO BHATTACHARYA, PAUL PFLEIDERER, and

STEPHEN A. ROSS*

ABSTRACTThe dichotomy between timing ability and the ability to select individual assets hasbeen widely used in discussing investment performance measurement. This paperdiscussesthe conceptualand econometricproblemsassociatedwith defining and meas-uring timing and selectivity. In defining these notions we attempt to capture theirintuitive interpretation.We offer two basic modelingapproaches,which we term theportfolioapproachand the factor approach.We show how the quality of timing andselectivity information an be identifiedstatistically n a numberof simple models, anddiscusssome of the econometric ssues associated with these models. In particular,asimple quadratic egression s shown to be valid in measuring iming information.

THE STUDY OF INVESTMENTperformancehas long sought to draw he distinctionbetween the ability to time he market and the ability to forecast the returnson individual assets.' This distinction has been viewed as a useful device forattributionof a manager'sperformance, .e., superiorperformance s due to eithertiming or selection ability or some combination of the two. Indeed, portfoliomanagersoften characterizethemselves as market timers or stock pickers. Anability to distinguish between these two sources of superior performance mayallow more accuratemeasures of the value of the servicesprovidedby managers.At the same time, the distinction between these two types of performancemayhave intrinsic interest. The distributionof prices and observed trades may bevery different depending on whether private information in the economy ispredominantlyabout market aggregatesor is predominantly irm-specific.The active nature of portfolio management is often based on the claim ofsuperior information. If this is so, then the main problem in performanceevaluation is inferringthe qualityof privateinformationpossessed by a portfoliomanager. Many tests do not successfully separate measures of quality frommeasuresof aggressiveness.By tradingaggressively,a managerwith little infor-mation may be able to mimic certain aspects of the performanceof a managerhavingmoreprecise information.Since a client can alwaysalter the aggressive-ness of a managedportfolio on his own account, aside frompossible savings intransactionscoststhat maymake mutual fundssuperior o individual nvestment,the valueaddedby a portfolio manager s a functionsolely of the precisionof his* StanfordUniversity,University of Californiaat Berkeley,StanfordUniversityand Yale Univer-sity, respectively.We would like to acknowledgethe useful comments of Mike Gibbons, MarkGrinblatt,DavidModest,and RobertVerrecchia.The third authorreceived upport rom he StanfordProgram n Finance.1A partial ist of referencesncludesTreynorand Mazuy[13],Fama[6],Jensen [8], Kon [9],KonandJen [10],Merton[12], Henrikssonand Merton[7], and ConnorandKorajczyk 3].

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716 The Journalof Financeinformation. Empirical tests of timing and selectivity should be designed torecoversomemeasureof the qualityof the informationa managerhas,as distinctfrom measuringthe investment strategies of the manager(such as the level ofchangesin the beta of his portfolio).We show that this can be done fora numberof interesting specifications.Relatedworkcan be foundin Admatiand Ross [2].Our focus in this paperis mainlyon the conceptualand econometricproblemsinvolved n distinguishing imingand selectionability.Althoughthe termstimingand selectivity have been used quite often, providingsatisfactorydefinitions isnot an easy task. Timing is often characterized(see, e.g., Jensen [8]) by theresponseof the informedmanagerto the private informationhe or she obtains.In the language we will use, there is a small set of portfolios, called timingportfolios,whichessentiallyprovide separating undsfor the managedportfolio-the change in the portfolio's composition in response to timing informationalways involves only shifting funds among the timing portfolios. Selectivity isusually thought of as the ability of a managerto pick individual assets. Twointuitivecharacterizationsof this notion are:(i) as distinct fromtiming, selectiv-ity information s statistically ndependentof the returnson the timing portfolios,2and (ii) different types or coordinates of selectivity information pertain todifferentassets. We first observethat these two characterizationsmay be incon-sistent with each other. Indeed, they are, strictly speaking, inconsistent witheach other if asset returns are normallydistributed,as we assume for most ofour analysis.3

In the next section we discuss two approachesto defining selectivity, whichare motivated by the observationabove.We call the first the portfolioapproach,and the second the factor approach.In the portfolio approachwe use the firstcharacterizationof selectivity. Specifically,for a given set of timing portfolios,selectivityinformationpertainsto the residuals n the regressionof asset returnson the returns of the timing portfolios.For example,if timing portfoliosare theriskless asset andthe marketportfolioof riskyassets, then selectivityinformationpertains to the residuals from the standard market-model regression. Thisdefinitionis consistent with many of the empiricalstudies in the literature,butit is somewhat awkwardand not easily motivated.In particular, t is difficult toreconcile the information structures that arise in this formulation with theinformationacquisitiontechnologythat gives rise to the existence of this infor-mation.In what we call the factor approach,a factor generatingprocess is postulatedforasset returns,andtiming and selectivity informationare interpreted n termsof their statistical relation to the factors and to the idiosyncraticterms in thegeneratingprocess. This approach is consistent with the above definition of

2This characterization as been important n the recent controversywith regard o the use of thesecurities market line to measureperformance(see Mayers and Rice [111and Dybvig and Ross[4, 51).It wasargued hat usingthe securitiesmarket ine is justified f thereis notiminginformation.Private information s assumed to be independentof the returns on the index. This is the mainmotivationbehind the portfolioapproachdevelopedbelow.'To see this most clearly,note that in this case, if there areL timingportfoliosand N assets, andif selectivity information is independentof the returns on the timing portfolios, then it can besummarizedby an N - L dimensionalsufficient statistic. It follows that if selectivityinformationsatisfies (i) above,then it is impossible hat there are N independent electivity signals, one for eachasset.



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On Timingand Selectivity 717timing, i.e., if informationpertainsto a set of K factors,then there are (at least)K timing portfolios.But now the selectivity information,which pertains to theidiosyncratic terms in the factor model, is truly information about individualassets that is statistically independent (given the factors) of returns on otherassets, and is, further, independentof the factor information that gives rise tothe timingportfolios.What is not true anymore s that selectivity information sdistinct from timing in the sense that this information is independent of thereturns on the timing portfolios. Thus, this definition captures the secondcharacterizationof selectivity mentioned above, but not the first. It is, in ouropinion,more naturaland easier to motivate.One result we derive, which is importantto the ability to study timing andselectivity empirically is that, surprisingly, he compositionof the timing port-folios obtainedin the factorapproach s independentof the quality of theprivateinformation.For example,if two managerseach have timing information (infor-mation that pertains to some K factors of the distributionof returns), then nomatterwhat the precision of the private informationis, the two managershavethe same timing portfolios, i.e., they shift the composition of their portfolioswithin the same set of separatingportfolios. This result, however,depends onthe managerpossessing no selectivity information.Informationabout the idio-syncraticterms affects the compositionof the timingportfolios,so that althoughthe responseto the timing information is such that L timing portfoliosareused,these portfolioswould be different for two managerspossessingselectivity infor-mationwherethe qualityof information is different.In particular, f the qualityandnature of private informationareunknown,then it is not possible to identifyex ante the timing portfolios and use them in a market-typeregression.Onemustinstead use the data to infer the identity of the timing portfolios.The aboveconceptualdistinctionsbetweentimingandselectivityarediscussedin Section I. We then examine the possibilities for attributingthe performanceof a managedfund to one activity or the other when only ex post returns areobserved.Contrary o Jensen [8], we can show that it is possible to disentangletiming and selectivity empirically.In Section II we discuss the statistical prop-erties that are special to the factorapproachwith regard o measuringperform-ance. In Section III we focuson the portfolioapproach,wherestatisticalmethodsformeasuringperformancemaybe different.The tests are essentiallyextensionsof the approach irst suggested n Treynorand Mazuy [13] anddeveloped urtherin Jensen [8]. Essentially it is assumed that the manager'soptimal portfolioposition is linearly related to his expectations of the returns to be earned onassets andtiming portfolios.This gives rise to an estimablerelationbetweenthereturns earned on the managedfund and a quadraticfunction of observables.Althoughthe factorapproachseems to be superiorto the portfolioapproachonconceptualgrounds,we find that it presents moredifficultiesin testing. SectionIV providesconcludingremarks.

I. The Distinction Between Timing and SelectivityIn this section we present a general definition of timing and then discuss twoapproaches o the definition of selectivity. The first approach,which we call the



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718 The Journalof Financeportfolioapproach, s implicit in much of the previousliteraturebut suffers fromsome conceptualproblems. The second, the factor approach, s not subject tothese problems and seems to us to be more in the spirit of the intuitiveinterpretationof these terms. In both approaches, he distinctionsareultimatelystatementsaboutthe type of information which managersreceive.For most of ouranalysis,we assumethat informationsignalsandasset returnsare distributed multivariate normal.4The information signals that managersreceiveare, if they are informative,correlatedwith some or all of the returns tobe realizedon the individualassets. Under what conditions can a signalbe calleda timing signal? In the approachtaken here, a signal is a timing signal if theappropriate esponseto any realizationof the signal involvesonly the shifting offunds among timingportfolios.Timing information s thereforecharacterizedbythe limited natureof the responseit produces.

To make this precise, assumethat there areN + 1 assets in the economyandthat the return on the ith asset is Ri. Assume also that a riskless asset exists,which is labeledas the O'thasset. (All of the conceptswe present here are easilyextended to settings in which no riskless asset exists.) A portfolioa, is an (N +1)-vectorwith ai equalto the weightplacedon asset i and Eo ai = 1. We assumethat the portfolio managerreceives a signal, Y, that is informative about thereturnsto be earnedon some or all of the N + 1 assets, and that this informa-tion is used to construct his or her optimal portfolio, a*(Y). Letlao, , ..., ao I be a set of N + 1 linearly independent portfolios. The set?a0, a?, . a.. Ih is a basis for R,N+1 and any portfolio a can be constructed bycombiningportfolios in the basis, i.e., a = nL0 yna for some yn Xn'=o Yn= 1.Thus, when the portfolio is chosen in responseto the information signal Y, wecan write, without loss of generality,a*(Y) = 'Yn(Y)a0. Let us consideraparticular set of basis portfolios and call the first L + 1 portfolios timingportfolios.Then, if the signal Y is a timing signal, the optimal portfolio a*(Y)can be constructed as a combination of the timing portfolios tao, ao, ... , a2j.

a*(Y) = Zi=o yj(Y)ai. (1)Upon receiving timing signals,a managershifts his funds amongthe first L +1 portfolios and never needs to choose portfoliosthat are not spannedby theseL + 1 funds. A simple example is markettiming, where only two portfolios areused,the riskless asset and the marketportfolioof risky assets, i.e.,

a,= a2 - (2)

where aoe s the value weight of the i'th risky asset in the portfolio of all riskyassets. Moregenerally,we mighthavetiming definedoverindustryportfolios.InIThis makesthe calculationof the optimalresponseto informationand the econometricanalysisin the next sectionmoretractable.



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On Timing and Selectivity 719this case a portfolioin la',, a2, ... , ao2Iwould be the value weighted portfolioofrisky assets corresponding o firms in a particular ndustry. There clearly are asmany ways to define specific timing abilities as there are ways to identify timingportfolios.Presumably, the timing distinction is meaningful only when L + 1,the number of timing portfolios, is substantiallyless than N + 1, the number ofassets. Note also that once the timing portfolios are specified, there are stillmany differentsignals that aretiming signals, i.e., that producea timing responsewith regard o these portfolios.Now considera managerwhose information s exclusively timing information.Let R be the vector of returns on the N + 1 assets. Then the return on themanager'sportfoliois

(a(Y))'R = Xi=oy1(Y)(a?)'R = Ei=o 'yi(Y)Rl, (3)whereR1-s the returnon the l'th timing portfolio.The timing information, Y, isvaluable if the i( Y) and the R??re correlated.If a time series of the yi(Y) andthe R1?s observable,one can in principledetect timing informationby lookingat the joint distribution between yl(Y) and R1?.Under the assumptionthat R,the vector of returns, s (unconditionalon any private information) ndependentlyandidentically distributedovertime, only an informedmanagercould choose Y1,(I = 0, ..., L) to be correlated with R?, 1= O, ..., L. In general we would likenot only to detect timing information but also to measure its precision. To dothis requires that we specify in some detail the manner in which the managerreacts to his information, i.e., the functional form of 'y(Y). Examplesare foundin the next two sections.In contrast to timing information,which pertains to portfoliosor aggregates,selectivity information is usually consideredto be information about individualassets. This implies that no restrictionscan be placed on the space spanned bythe optimal portfoliosof a managerreceiving selectivity information.Indeed,fora managerwho receives selectivity informationabout each individualasset, theset of all possible portfolio responses, ta*( Y)}, spans RN. Thus it appears'hatwe cannot easily define selectivity in terms of the portfolio space in which themanagerreacts to his information.In the end, signals about the L + 1 timingportfoliosand about the N + 1 assets are meldedtogetherto produceat most an(N + 1)-dimensionalresponse.Howthen do we define selectivity informationonce we have specifiedwhat wemeanby timing? We discuss here two approaches o this problem.To illustratethese two we considera very simple example.Assume there are three assets (N= 2) with asset 0 being riskless. Let asset 1 and asset 2 be riskywith identicallyand normally distributed returns (unconditionalon any' private information).The followingthree portfolios span the portfolio space.

a 0) al 1/2\ (a0\ (4)0 1~~~~~/2

Let us define timing in terms of the first two portfolios in the basis (i.e., theriskless asset and the equally-weightedportfolioof risky assets). Thus a manager



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720 The Journal of Financewho receivesonly timing information has an optimal portfolio of the form

11 - (k)a*(Y) = 1/2'y(Y) . (5)1/2'y(Y) IIn the portfolio approach o distinguishingselectivity fromtiming we define aselectivity signal to be one that is (potentially)informative about the returnsonsome of the assets but uninformativeabout the returnson the timing portfolios.If YRSs selectivity information, then E(R?I Ys) = E(R? ) for 1= 0, 1, 2, ... , L orin the context of our particularexample, E('/2(Rl + R2) I Y5S) - E('/2(Rl + R2)).To see what form such a signal might take,considerourexample.We can imagineregressing he returns of the secondand third assets on the returns of the secondportfolioin the basis. We have

Ri = E(R,) - oV(1R, 1) (1?) + Va(1R ) R1 + vVar(R ~R) Var(Ri)-Cov(R2,R,) COV(R2,RM) ~R2 = E(R2) Var(R) E(R,1)+ Var(R?) R, + ^2 (6)

where i, and i2 are independentof R?. Informationabout the realizations of i,or i2 is therefore informative about the returns on the individual assets butrevealsnothing about the return to be earnedon the timing portfolio,Rl. Note,however, that since ?/2(R1+ R2) = R?, ?/2(i + V2) must equal zero for allrealizationsof i, and i2. This means that i, and i2 must be perfectlycorrelated.Clearlyit is impossibleto have independent information about R,, R2, and R1.Selectivity information,when it is defined in this way, cannot be informationexclusivelyabouteach of the N individualriskyassets. It must alsobe noted thatthe randomvariables, i, and i2, do not have a natural interpretation in thisapproach.That is, what does it mean for a manager o gatherinformationaboutR1and R2 which happensto be uninformative aboutthe return on the portfolio,1/2(Rl + R2)? We do not have an appealing story about the source of thisinformation.Thus, the approachseems artificial.Nevertheless,this approach odefining selectivity is consistent with much of the literature about timing andselectivity. For example, in many empiricaltests designedto detect and distin-guish timing and selectivity, portfolio returns are regressedon a constant termand terms which control for timing, i.e., which are a proxyfor the interaction of,y((Y)and 1R?. t is claimedthat the constant term in these regressions s relatedto the selectivity performanceof the portfolio manager.This will be true if wedefine selectivity in the mannerjust proposed,as will be shown in Section III.The optimal portfolio, a*(Y), of a manager having only selectivity informationis independentof the returnson the timing portfolios,R1.Thus in the regressiondescribed,only the constant term can be affectedby the manager'sactions.The secondapproach o defining selectivity is, we believe,morein the spiritofthe intuitive interpretationof this term and the concept of timing. We call thisthe factor approach.The terms macro-forecastingand micro-forecastingareoften used in place of timing and selectivity. These suggestthat timing involvespredictingthose factors that affect the aggregate,while selectivity is based on



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On Timing and Selectivity 721forecasting firm-specific determinants of asset returns. Assume that in ourexamplethe returnson assets 1and2 can be represented n termsof the followingvery simple factormodel5:

R, = E + 6 + e1 (7)R2= E + 6 + E2,with e, and e2independentlyand identicallydistributed.In this factor model, itis natural to consider informationabout 6 to be timing informationand infor-mation about e, and e2to be stock specific or selectivity information.Note thata managerobservingonly informationabout 6 will indeedformportfolios spannedby ao and ao in (4). The managerwill engageonly in timing. In contrast to thefirst approach,however,a managerobservingselectivity information, .e., infor-mationabout either e, or e2,does learnsomethingabout the returnon the timingportfolio,R?.Thus, the portfolioselectionsof a managerreceivingonly selectivityinformation may be weakly correlatedwith the return on the timing portfolio,Ri. They are, however, ndependentof the realization of the factor,6.Empiricaltests designed to detect timing ability as defined in the factor approachwilltherefore ook for evidenceof correlationbetweena manager'sportfoliopositionsand the factorrealizations (ratherthan with returnson timing portfolios).In a model with more than one factor, it is natural to define several timingportfolios,one for each factor.A managerwith only timing informationwill shifthis funds only amongthese factor-timingportfolios.Even when there are severalfactors,it is still possibleto defineN distinct selectivity signals corresponding othe N asset-specific disturbances, ss, i = 1, 2, . .. , N.The factormodel resolvesseveralconceptualproblems nherent in the portfolioapproach.First, selectivity informationcan be truly asset specific. In the factorapproach, here are as many degreesof freedomto define selectivity signals asthere are riskyassets. This is not so in the portfoliodefinition.Second,andmoreimportantly,in the factor approachthe informationsignals potentially corre-spondto economicallymeaningfulsourcesof uncertainty.As arguedabove,thereis no satisfactoryway to give such an interpretationto the selectivity signals inthe portfoliodefinition.Note that, underjoint normality,with N risky assets there always exists asufficientstatistic whose dimensionis at most N. A managermay observe (underthe factor definition) K timing signals and N asset-specific selectivity signalsbut these can be collapsed nto N signals.The same is true for managersreceivingtiming and selectivity signals under the portfoliointerpretation.In other words,any jointly normalsignal can be written as Y = C R + j for some matrix, C. Itthus seems that the only distinctions we can make between different types ofinformationsignalscan concerneither the matrix, C, or the variance-covariancematrix of the errors, i1.Nevertheless, the two approachesdiscussed above aredistinct in some importantways. First, the distinction on the individual levelcomes in when we consider statistical tests of performance,since the set ofobservablesand the statistical hypotheses that are tested wouldbe different in

'We choosehere a particularly implerepresentationof a factor model to illustrate he approach.Obviouslyhe definition canbe applied o any factormodel. This is done in the next section.



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722 TheJournalof Financethe two formulations.Second,the portfolioandfactor specificationsaredifferenton the equilibrium evel. If, for example, we postulate that a large numberofinvestorscollect timing and selectivity signals of some specific nature, and thatthe private informationis diversein the sense that individual signals are condi-tionally independent of each other (give some sufficient statistic), then theaggregationof these diverse pieces of information n asset equilibriumpricesmaygive rise to observationallydistinct economies.6

II. A factor model of timing and selectivityIn this section we develop an explicit model of timing and selectivity underthefactor approach.This requiresthat we specify the nature of the informationreceivedby a portfoliomanager.We aremainlyinterested in determiningwhetherit is possible to identify empiricallythe quality of a particularmanager's imingand selectivity information roma time series of the returnsearnedon a managedportfolio.We showthat in principlethis is always possiblebut in practice it maynot be feasible.7This leads us to consider in the next section a simpler case inwhich the problemis to determine the quality of only the timing information.Ratherthan measuring he qualityof the selectivity information,we simplytestto see if such information is present. These more modest goals seem to beachievablewith the limiteddataon mutual fund returns that is usually available.Assume that the N risky asset returns are generatedby a factor model. Thereturns earnedin periodt are given byRt = E + Bbt + (t~ (8)whereRtis the vector of the N riskyasset returns,at is a K-dimensionalrandomvector, (t is an N-dimensional randomvector, and B is an N x K dimensionalmatrixof the factor loadings.We can normalizethe factors so that Var(St)= Iwhere I is the K dimensional identity matrix. Let D be the variance of (t andassume that at and (t areuncorrelatedand distributedmultivariatenormal. Sincewe do not requirethat D be diagonal, (8) is completelygeneral.Of course,if wewant to think of the at as being meaningfulfactors,then we will want to assumethat D is nearly diagonalso that most sourcesof common variation are due tothe at terms.There are two types of signals in our model. Each signal is assumed to bejointly normallydistributedwith at and ft. Timing signals have the form Y5Tat + it; correspondingly,selectivity signals are written as Yt = it + Ot. It isassumedthat the vectors -t and Otare uncorrelatedand that each is independentof at and (t. The assumptionthat jtand Otare independentis made to simplifythe analysis. Since q-t orresponds o the errorsmade in forecastingthe factorsand Ot s relatedto the errors madein forecastingthe firm-specificdisturbances,it seems sensible to assume that they are independent.We let Var(it) = 2 and

6To see this, one needs to examine the measurability equirementof prices.When the aggregateof privateinformations observationallydistinct,then the equilibrium esultswouldalso be distinct.This argument s similarto that in Admati [1], section6.'See Admatiand Ross [2] forrelatedresultson the identifiabilityof performanceparametersn asimilarmodel.



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On Timing and Selectivity 723Var(dt)= Qand assume that these are both positive definite. The quality of amanager's nformationis completelydeterminedby 2 and Qand it is these twovariancematricesthat we wish to identify.It is clearly not possible to recover 2 and Q from a time series of portfolioreturnsunless some restrictions are placedon the way the managerrespondstohis information. One of the simplest assumptionsspecifies a linear responsetoforecasts.This is consistent with the manageracting as if he were maximizingthe expectedutility of someonewith constant absolute risk tolerance.With thisassumption,the optimalportfolioof riskyassets is

*(Yt, Yt) = p(Var(RtI Yt, YYt))(E(RtI Yt, ) - Roe), (9)wherep is a measureat risk tolerance,E(RtIYt, Ys) is the vectorof conditionalexpected returnsgiven the manager's nformation,and e is the vector of ones. Itis easily shown,using the theory of normalvariables,that

E(Rt YT, Yt) = E + B(I + 2)'(bt + it) + (D + Q) D(et + at), (10)and

Var(RtI YT, Yt) = B(I + 2-1)-'B' + (D-1 + -1)-1. (11)Let E = E - Roe be the expectedexcess returnvector.Then the optimalportfolio,*(5]T, 51) isp(B(I + 2-1)-1B' + (D-1 + Q-1)-1)-'(E + B(I + 2)-1(6t + it)+ (D + Q)-1D(et+ at)) (12)Firstnote that the informationsignal, YT, s trulya timingsignal.The portfolioposition of a managerreceiving only timing information is

a*(YT) = p(B(I + 2 -')-'B' + D)'-(E + B(I + 2 )-'(6t + )). (13)Thus the response to the information signal, at + it, is simply to shift fundsamongK timingportfoliosin the span of

(B (I + 2 -1)-1B' + D) -1B (I + 2) -1. (14)From (14) it might appearthat the set of timing portfoliosdependsupon 2, thenature of the timing informationreceived.This is not true. It is shown in theappendixthat the span of (B (I + 2 -1)-1B' + D) -1B(I + 2 )-1 is independentof2. Thus all managerswhoreceiveonly timing informationshift fundsamongthesame set of timing portfolios.If a fund managerreceivesboth timing and selectivity information, it is stilltruethat the responsesto timing informationarerestricted o a set ofKportfolios.However,unlike the situation when only timing information s received, t is nottrue that this set of portfolios is independentof the quality of the informationreceived.The timing responseportfoliosnow lie in the span of

(B(I + 2-1)-1B' + (D-1 + Q-1)-1)-1B(I + 2)-1, (15)which does dependon the value of Q, the qualityof the selectivity information.Thus two managerswho receive the same timing signals but selectivity infor-



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724 The Journal of Financemation of differing qualities will respond to the timing signals in differentfashions.This makes the disentanglementof timing and selectivity much morecumbersome han it wouldotherwisebe.Despite the fact that the timing portfolios depend upon the quality of theselectivity information,it is in principlepossible to identify both z and Q,evenif the parameterp, which measuresthe aggressivenessof the manager'sresponseto information, s unknown.Using (12) it can be shownthat the realized excessreturn on the managedportfoliois

-P a*( YT, 1S)'(1t - Roe)= p[E'-WE + E -1B(I + 2)-(6t + 1t)

+ E' P1(D + Q)-'D(et + Ot)+ 6b'B' -1E + b6'B' -JB(I + b)l(6t + 7)t+ tB J-1(D + Q)-1D((t + at)+ 'T 1E + Et J-1B(I + Y)(-10 ht)+ 'TI-'(D + Q)-1D(t + r)], (16)whereT = Var(Rt YT, Ys) = B(I + 7'1B' + (D-1 + Q-1)-1.If one assumesthat E, B and D are known parametersof the generatingprocess, then it is

possibleto determine(expost) the realizationsof at andZt.The factorrealizationscan be obtained by the cross sectional regression of Rt - E on B. If N issufficiently large (and the elements of D are bounded),at can be recoveredwithnegligible error. The idiosyncratic terms are then recovered: -t= Rt- Bt.Consider now the time series regression of Rt on a constant, Zt, , and thedistinct terms in the matrices tb' (K(K + 1)/2 terms), - (N(N + 1)/2 terms),and `tV&(NKerms). The constant term is easily shown to be a consistentestimatorof pE'T 'E. For the other terms we haveplim(coefficientson at) = pE' -'B(I + (I + 2)-1) (17)plim(coefficientson (t) = pE'-1'(I + (D + Q)-1D) (18)

plim(coefficientson btS) = p(sym(B-' 1B (I + 2 ) 1)) (19)plim(coefficientson tt) = p(sym('W1(D + Q) 1D)) (20)plim(coefficientson -t[) = p(*-1B(I + 2)-l + D(D + Q)-1x-1B), (21)

wheresym(A) refers to the M(M + 1)/2 distinct terms in the symmetricversion8of the M x M matrixA.If we include the constant term,we have a total of (K2+ 3K)/2 + (N2 + 3N)/2 + NK + 1 estimatedcoefficients.The numberof unknownparameters n I, Q,and p is (K2+ K)/2 + (N2 + N)/2 + 1. It appearsthat we potentially have K +N + NK overidentifyingrestrictions.In fact there aremanymore than this. Firstthere are the restrictionsrequiredfor 2 and Qto be positive definite. Addedto'The symmetricversion of a squarematrixA is (A+ A')/2.



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On Timingand Selectivity 725these are restrictions related to the information contained in the disturbanceterms in the proposed regression. These terms are heteroskedasticwith thevariancerelatedin a knownwayto p, 2 and Q.9Given the nonlinearities nvolvedand the distributional assumptions made, an attractive estimation procedurewouldbe the maximum likelihoodapproachwhich explicitly accounts for theseadditionalrestrictions.Whether this procedure s used or not, it is possible torecoverthe appropriatemeasures of the quality of both the timing informationand the selectivityinformation.Note also that to identifythe informationqualityof a manager using the procedure ust discussed, it is not necessaryto assumethat any particularmodel of asset pricingholds.Althoughboth 2 and Q(as well as p) can in principlebe recovered, n practicethis may generally prove too difficult or impossible. The reason is that thenumberof regressors n (17)-(21) is extremely largeand most likely exceedsthenumber of time series observations obtainablefor any fund. In the next sectionwe look at a much simplerproblem based on the portfolio approachand showthat recovery of at least the quality of timing information will generally befeasible.

III. A Portfolio Model of Timing and SelectivityIn the last section it was shown that when the factorapproach s used to definetiming and selectivity, it is possible to identify both the quality of timinginformation and the quality of selectivity information.This can be done even ifthe only observableeffect of the manager'sinformation in each period is thetotal return earned on the managed fund; for identification of the quality ofinformation it is not, for example, necessary to observe portfolio positions.Unfortunately,because of the large number of interactions between informationsignalsand asset returns, it was necessary to include an extremelylarge numberof regressorsin the regressionproposed in the last section. For this reason,estimation of the quality parameters of the timing and selectivity informationseems to be possible under the assumptions of the last section only when thenumberof time series observations is impractically arge.In this section we pursue the more modest goal of estimating the quality ofonly the timing information.The testing procedureproposedwill only be capableof determiningwhetherselectivity information s or is not present; the quality ofthe selectivity information will not be identified. By reducing the number ofparameters to be estimated, we obviously simplify the estimation problem. Itshould be noted, however, that the complexity of the estimation procedurediscussed in the last section is due to more than just the large number ofparameters hat must be estimated. Much of the difficulty can be attributedtothe fact that underthe factordefinitionthe timing responseportfolioscannot beidentifiedwithout knowingthe quality of the selectivity information.Althoughmanagersreceiving only timing informationrespondto their informationusing

9The error term is p(Et'*-1B(I + 2) -'t + 4' `1(D + Q) `D&t + StB I'1B(I + 1)-'it +atB I*(D + 9)-rD#t + (t *-J1B(I+ 2)-'it + (I '1(D + )-1DDt).



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726 TheJournalof Financethe same set of timing portfolios, this is not the case for managerswho receiveboth timing and selectivity information.(The span of (15) is not independentofU.) Becausethe set of portfoliosused to respondto factorinformationcannot bedefinedwithout knowledge of the quality of selectivity information, the estima-tion of informationqualitybecomesmuch morecomplexthan it wouldotherwisebe.In contrast to the factor approach,the portfolio approachto defining timingand selectivity does allow one to specify the timing response portfolioswithoutknowingthe quality of the selectivity information. This is because selectivityinformationunderthe portfolioapproach s by definition completely uninform-ative about the returns to be earned on the timing portfolios. The estimationproblem s thus simplified if selectivity information s assumedto be independentof the return earned on the timing portfolios.Despite the conceptualproblemsinherent in the portfolioapproach,or perhaps becauseof them, identificationofinformation quality is, given certain assumptions, easier under this approachthan it is under the factorapproach.To show howestimation of the qualityof timing informationcan be accom-plished under the portfolio approachwe consider a simple case where there isonly one timing portfolio. Let there be N risky assets and let the variance oftheir returnsbe V. The timing portfoliocan in principlebe any portfolioof theN risky assets. In almost all of the previousdiscussions of timingin the literature,the timing portfoliois assumed to be the marketportfolio.We will not makethisparticularassumption here but the reader is free to think of the timing beingdiscussedas markettiming. Let the timing portfoliobe a T and let its variancebe S2 = (aT) 'VaT. It is possible to defineN-1 portfolios, aS, j =1, .. ., N-1,such that

(aoe)'e= 1(aT) Vajs 0 (22)

for eachj. We can defineN new assets with the first being the timing portfolioand the remainingN - 1 being a set of portfolios which satisfy (22). In otherwords we can rotate assets so that the first is the timing portfolio and all theother portfolios are uncorrelatedwith the first. Let RT be the return on thetiming portfolioand Ri, be the return on the portfoliowith weights a'S.Then

(ATVar Ri. a= ON1 ) (23)

RN-l,twhereON-1 is the N - 1 vector of zeroes and V is the variance-covariancematrixof the N - 1 portfoliosthat have returnsorthogonalto the return of the timingportfolio.In the portfolioapproachwe define the timing signal to be Yt =AT+nt and the N - 1 possible selectivity signals to be of the form Yis,t R't + Oi,t.We assumeas before that returnsandthe informationerrors -t and at arejointlynormally distributed and that it and 0atare independent. The parameters of



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On Timing and Selectivity 727interest are again Var(it) = a2 and Var(#t)= Q.We will be able to estimate a2but given our more limited goals we will only test whether Q is finite or not, i.e.,whether selectivity information exists or not.As in the previous section it is necessary to specify how the managerreacts tohis information. Again we make the assumption that the response is linear andtherefore consistent with maximization of utility with constant absolute riskaversion. The position taken in the N portfolios by the manager s given by

~~~~~~~~~~2ak\ONl +a ER +R

(a2 + a2 a;- a20 ERT + U2 ytT \O-1 (V-1+ Q_1)_ \ IN-,1-(V + Q) -V)E(Rt) /+ (V+ )-1Vs -Roe

whereRo s the riskless rate.The returnrealizedon the managedportfolio in period t, Rt', is consequentlyequal toE(R-t - Ro Ro T + P (RT)2 + P i7t t (25)

whereut = p(1 )'(V1 + Q-1)((IN1 - (V + 9)-'V)E(RJ)+ (V + Q)`VYs - Ro) (26)ad (R-st)'=(Rsit R-s it).Now consider the regressionof the managed portfolio's return on a constantterm, the return on the timing portfolio and the squared return on the timingportfolio:

RtI = ro + r1Rt' + r2(R + ?t (27)Since utand -tareby assumption independentof Rt , r2 is a consistent estimatorof p/U2. As we argued above,the parameterof interest is U2. We need thereforeto get an estimate of p. This can be done using the residualsof the regression.The true disturbanceterms are heteroskedastic and equal to (p/u7)iItRT i,-E(ut). Considerregressingct2on a constant and the squareof RI'.

t = Ao + A(RT)2 + t (28)Under the distributionalassumptionswe have made,plim(Al) = p2/U2. Since wehaveestimates f p/U2 andp2/v2 , wecanrecover oth a2andp.The two step procedure just discussed clearly does not produce the mostefficient estimates of the information quality parameter, a.. More efficientestimates can be obtainedby taking into account the heteroskedasticityof thedisturbanceterms. One possible approachwould use the estimates of p and a.obtained in the inefficient mannerjust describedto predict the varianceof theresiduals.A GLS procedurecouldthen be used to obtain better estimates of thecoefficients in the two regressions.As before, a maximumlikelihoodapproachwhich completely accounts for the structure of the model is probably most



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728 TheJournalof Financeappropriate. It is beyond the scope of this paper to discuss in detail suchrefinements.Havingshown that the qualityof timing informationcan be identified,we nowturn to the detection of selectivity information.Responses to selectivity infor-mation affect only the distributionof uz,n (25) and ro can be shown to be aconsistent estimatorof E(ix).It is not possible, however,to infer anythingfromthe value of E(at) unless further assumptions are made about asset pricing.Assume,for example,that the CAPM holds and RtI is the return on the marketportfolio.The portfolios a'Sare constructed to have returns orthogonalto Rt .This means that E(R,t) = Ro,the riskless rate, for all i. A managerwho does nothave any selectivity information will not be able (in expectation) to earn morethan the riskless rate by shifting funds among the portfolios a'S.If a managerdoes have valuableselectivity information(finite Q)and if he respondslinearlyto this informationas is assumedin (26), then he will earn (in expectation)morethan the riskless rate and this will be reflected in a highervalue of E(ut). It isclear that we cannot learn from E(ut) all that we need to know to value themanager'sselectivity information. The value of the selectivity informationde-pends on the matrix,Q,and is not summarized n the scalar,E(at). Nevertheless,it is still encouraging hat the simple regressionproceduredescribed above canbe used to identify fully the quality of timing information and to detect theexistence of selectivity information.

IV. ConclusionsThe importantproblem in performanceevaluation is the measurement of thequalityof the informationpossessedby a portfoliomanager.Many tests proposedin the literature do not successfully separate the aggressiveness of a managerfrom the quality of the information he possesses. To measure the quality ofinformation t is necessaryto specifyboth the nature of the informationreceivedand the nature of the manager'sreaction to it. For this the distinction betweentiming informationand selectivity informationmay be useful. We have arguedthat the distinctionbetweentiming and selectivity is not straightforward ndwehave examinedtwo approaches hat can be used to makethe distinction precise.Underthe factor approachtiming informationis informationabout the realiza-tions of factors which affect the returns of many assets. Selectivity informationis then defined to be informationabout the asset-specificdeterminants of assetreturns.Under the portfolioapproach, iming information is information aboutthe returns to be earned on a pre-specifiedset of timing portfolios. Selectivityinformationunderthis definition is any information hat is uninformativeaboutthe timing portfolio returns but informativeabout some asset returns.

These two approachesto defining timing and selectivity are not equivalent.They are observationallydistinct both in their implicationsfor the joint distri-bution of returns on assets and managedfunds and in their implicationsforassetpricing.We have shown that under each interpretation t is possible to identifyempiricallysome measure of the quality of information possessed by a fundmanager.This is possible even when all that is observedare the ex-post returnsearned on the managedfund. It is important to note, however,that the specifi-



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On Timingand Selectivity 729cation of these tests depends in critical ways on the approachtaken to definetiming and selectivity. Althoughthe factor approachhas conceptualadvantagesover the portfolio approach,testing may be more feasible if the informationreceivedby managers s consistent with the portfolioapproach.

It should be noted finally that our discussion has abstractedfroman equilib-riummodelexplainingasset returns.In discussingempirical ests we have madeassumptionsof independence(and normaldistributions) hat are consistent withmost of the empirical literature. However, such assumptions have not beenjustified by a theoretical intertemporal asset pricing model which explicitlyadmits asymmetric nformation. If returnsare determined n a worldof homog-enous beliefs,onlyan insignificantportionof the marketcan possess any superiorinformation. The importance of measuring performance in such a world isquestionable.Appendix

In this appendixwe show that in the factor model developedin section II, thefactor timing portfoliosused by managersreceivingtiming informationdo notdepend upon the quality or nature of that information.This requiresthat weshowthat the span of the matrix(B (I + 2-1) -1B + D) -1B (I + 2 )-1 (29)

does not dependon Z. To do this we will showthatspan[(B(I + 2-1)-1B' + D)-1B(I + 2)-1] = span[(BB' + D)Y1B] (30)

for all Z. Let F and G be two N x K matrices with N > K. Then span(F) =span(G) if and only if there exists a K x K matrix H such that F = GH. Since(I + 2) is invertible, (30) is established if it is shown that there exists H thatsolves(B(I + 2 -1)-B' + D) 1BH= (BB' + D)1B. (31)

Let Q = (BB' + D) 1B. Then B = (BB' + D)Q orB - BB'Q = DQ. SubstitutingQ into (31) and rearranginggives us BH - B(I + 2-1)-1B'Q = DQ. Combiningthese two we obtainB(I - B'Q) = B(H - (I + 2 -1)-1B'Q), (32)

which means that H must satisfyI - B'Q = H - (I + 2-1) -B'Q. (33)

A matrixH which solves (31) thereforeexists and is givenbyH = I - (I - (I + 2-1)1-)B'Q= I - (I - (I + 2 -1)-1)B t(BB ' + D) 1B. (34)

Since all matrices (B (I + 2-1)-B' + DJ)1B have the same span as (BB' +D) 1B andthe spanningrelationdefines equivalenceclasses,we haveprovedtheresult.Q.E.D.



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730 The Journalof FinanceREFERENCES

1. Anat R. Admati. ANoisy RationalExpectationsEquilibrium or Multi-AssetSecuritiesMar-kets. Econometrica3 (May1985),629-657.2. Anat R. Admati and Stephen A. Ross. MeasuringInvestment Performance n a RationalExpectationsEquilibriumModel. Journalof Business58 (January1985),1-26.3. GregoryConnorand RobertA.Korajczyk. PerformanceMeasurementwiththe ArbitragePricingTheory:A New FrameworkorAnalysis, Unpublishedmanuscript,NorthwesternUniversity,July 1984.4. PhilipH. DybvigandStephenA. Ross. DifferentialnformationandPerformanceMeasurementUsinga SecurityMarketLine. JournalofFinance40 (June 1985),383-399.5. - . TheAnalyticsof PerformanceMeasurementUsing a SecurityMarketLine. JournalofFinance40 (June1985),401-415.6. Eugene Fama. Components f InvestmentPerformance. ournalof Finance27 (June 1972),551-568.7. RoyD. Henriksson,andRobertC. Merton. OnMarketTimingand InvestmentPerformanceI:Statistical Proceduresfor EvaluatingForecastingSkills. Journal of Business 54 (October1981),513-533.8. MichaelJensen. OptimalUtilization of MarketForecastsand the Evaluationof InvestmentPerformance. n SzegoandShell,MathematicalMethods n InvestmentandFinance.Amster-dam:North Holland/AmericanElsevier,1972.9. StanleyKon. The MarketTimingPerformance f MutualFund Managers. ournalofBusiness56 (July1983),323-347.10. Stanley Kon and Frank Jen. The InvestmentPerformanceof MutualFunds:An EmpiricalInvestigationof Timing, Selectivity and Market Efficiency. Journal of Business 52 (April1979),263-289.11. DavidMayersand EdwardRice. MeasuringPortfolioPerformanceand the EmpiricalContentof Asset PricingModels. Journal ofFinancialEconomics7 (March1979),3-28.12. RobertC. Merton. OnMarketTiming and InvestmentPerformancePart I: An EquilibriumTheoryof Valuefor MarketForecasts. ournal of Business54 (July 1981),363-406.13. Jack L. Treynorand F. Mazuy. Can MutualFunds Outguessthe Market. HarvardBusinessReview44 (July-August1966),131-136.

DISCUSSION

ROBERTE. VERRECCHIA*:hroughoutthe portfolio performanceevaluationliterature,attempts are made to distinguishbetween markettiming ability andforecastingability (as it relates to the returns on individualassets). But theseattemptshave not been altogethersatisfactory,in largepart becausethe notionsof timing and selectivity hat have been introduced nto the literaturehaveusuallybeen defined in terms of their regressioncharacteristics,as opposedtoprimitiveeconomicelements (see, for example,Grinblatt-Titman[1], which alsoprovidesan excellent synthesis of the relevantliterature).To the extent that thepaper on timing and selectivity by Admati, Bhattacharya,Pfleiderer,and Ross(ABPRhenceforth)grappleswith variousinterpretationsof these phenomena, tfills an importantvoid in the literature.The controversyis whetherone shouldbe encouragedor discouragedabout the possibility of identifying timing infor-mationand detectingthe existence of selectivity informationon the basis of theevidenceput forth in the ABPR analysis.

*The WhartonSchool,Universityof Pennsylvania.

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Admati Et Al 86 on Timing and Selectivity