The Timing of Analysts Earnings Forecasts 1

Transcription

1 The Tmng of Analysts Earnngs Forecasts Ilan Guttman Stanford Unversty Graduate School of Busness 58 Memoral Way Stanford, CA May 8, 005 I am grateful to Eugene Kandel for hs gudance and encouragement. I would also lke to thank Sasson Bar-Yosef, Elchanan Ben-Porath, Ohad Kadan, Motty Perry, Madhav Rajan, and semnar partcpants at Columba Unversty, Hebrew Unversty, Northwestern Unversty, Stanford Unversty, Tel Avv Unversty, and Tlburg Unversty for ther helpful comments and suggestons.

2 Abstract Most of the lterature assumes that the order and tmng of analysts earnngs forecasts are determned exogenously. Ths paper analyzes the equlbrum tmng strateges for analysts. Consstent wth the pror lterature I assume that analysts care foremost about the accuracy of ther forecasts, but n some cases may have an ncentve to bas ther forecasts. The paper further assumes that nvestors reward early forecastng analysts. The man trade-o that an analyst faces n determnng the tmng of hs forecast s between precson and tmelness. The paper ntroduces a tmng game wth two analysts, derves ts unque pure strateges Subgame Perfect Equlbrum, and provdes emprcal predctons. The equlbrum has two patterns: ether the tmes of the analysts forecasts cluster, or there s a separaton n tme of the forecasts. All else equal, an ncrease n the precson of an analyst s prvate sgnal nduces earler forecast by ths analyst, and ncreases the lkelhood that the analysts forecasts wll cluster n tme. Ths predcton may be used n emprcal studes to nfer the precson of the analysts sgnals from the observed tmng of the forecasts.

3 Introducton Sell-sde analysts are one of the most mportant sources of nformaton for nvestors n the stock market. Among other servces, they provde early forecasts of rm s earnngs, forecasts whch nvestors use for stock valuaton. The analysts forecasts are based on nformaton they generate prvately as well as on publcly avalable nformaton, whch ncludes pror forecasts of other analysts. Ths suggests that every analyst s both a suppler of nformaton to other analysts and a consumer of such nformaton, that comes from other analysts. The degree to whch an analyst plays each role s determned by hs poston n the sequence of forecasts announcements that s to say the tmng of hs forecast. The queston that generated ths paper s whether the tmng of analysts forecasts s determned exogenously or at random, as mplctly assumed n much of the lterature, or whether analysts choose the tmng of ther forecasts strategcally. The answer to ths queston may yeld addtonal nsghts nto the behavor of sell-sde analysts that receved so much attenton recently. Spec cally, the nformaton contaned n the tmng and order of analysts forecasts may help decpher ther nformatonal content. Ignorng ths nformaton may lead to nconsstent nferences. In ths paper I propose a theory for the tmng of analysts earnngs forecasts, and analyze ther equlbrum tmng and reportng strateges. I follow the lterature n assumng that analysts care prmarly about the accuracy of ther forecasts (e.g. Mkhal, Walther, Wlls [999], Hong and Kubk [003]), but n some cases may also have an ncentve to bas ther forecasts (e.g. Dugar and Nathan [995], Hong and Kubk [003], Lm [00], and Das et al [988]). Jackson (005) documents the analyst s trade o between accuracy and optmstc bas. In a setup where the analysts objectve functon s based only on these two assumptons, all the analysts would optmally make ther forecasts mmedately before the earnngs announcement by the rm, snce ths s when ther forecasts are the most accurate. Ths s clearly not a reasonable outcome. Investors would be wllng to reward a devatng analyst who provdes an earler sgnal. Hence, there must be an o settng e ect. To capture ths e ect, I propose an addtonal component to the analyst s payo functon. I assume that the compensaton of an analyst declnes n the precson of the nvestors belefs about the earnngs of the rm at the tme of hs forecast. Thus, the expected payo of an In hs ntroducton, Jackson (005) says: the analyst must trade o the short-term ncentve to le and generate more trade aganst the long-term gans from buldng a good reputaton.

4 early-reportng analyst s hgher than the expected payo of an analyst who s makng the same forecast at a later tme. Ths seems to be a natural assumpton. Unnformed nvestors are wllng to pay for nformaton (forecast) that ncreases the precson of ther belefs. At the extreme case where nvestors are perfectly nformed about the forthcomng earnngs of the rm, a forecast s worthless to them, and obvously they are not wllng to pay for addtonal nformaton. Ths new assumpton nds an emprcal support n Cooper, Day and Lews (00) and n Jackson (005). Followng the ncorporaton of ths new assumpton about the analyst s payo functon, the analyst faces the trade-o between an earler, but less precse forecast, and a later but more precse one. I further assume that a contnuous stream of publc nformaton from exogenous sources (other than analysts earnngs forecasts) arrves over tme. 3 As to nvestors, I assume that they (as well as other analysts) do not necessarly know the actual bas of an analyst, whch further complcates ther nference. I start by dervng and analyzng the optmum n a sngle analyst case (hereafter the unconstraned optmum), whch provdes the basc ntuton and serves as a benchmark. The optmal forecast tmng for a sngle analyst s determned by the precson of hs prvate sgnal and by hs cost of a forecast error. The ntuton s straghtforward: hgher precson nduces an earler forecast, whle the hgher cost of an error postpones the forecast for the purpose of ganng more nformaton over tme. Next, I ntroduce a tmng game wth two analysts. The game has a unque Subgame Perfect Equlbrum n pure strateges. The equlbrum takes one of two possble patterns. When the two analysts are su cently d erent from each other, then each publshes hs forecast at the respectve unconstraned optmal tme the non-clusterng pattern. The only other alternatve, s when the two analysts ssue forecasts one mmedately after the other, creatng an endogenous clusterng n tme of forecasts. The lkelhood of the clusterng equlbrum pattern declnes n the dstance between the unconstraned optmal tmng of the two analysts; ncreases n the precson of the prvate sgnals of the analysts; and ncreases n In ther ntroducton, Cooper, Day and Lews (00) state: Snce brokerage rms pro ts depend drectly on commsson revenues, analysts compensaton s based, n part, on the tradng volume generated by ther research. Ths gves superor analysts an ncentve to release nformaton before other analysts n order to capture tradng volume for ther rms. Jackson (005) shows that hgh reputaton analysts generate more trade to ther brokerage rm, and that tmelness may be sgn cantly more mportant than accuracy n determnng an analyst s rankng. 3 Ivkovc and Jagadeesh (004) nd that the nformatonal content of analysts earnngs forecast revsons, generally ncreases over event tme.

5 the precson of the nvestors belefs about the bas of the analysts. At tmes of extensve arrval of new nformaton (around an exogenous event), the model predcts an endogenous clusterng of tmngs of analysts forecasts that are very common n the data. The model predcts the order, the tmng, and the reported forecasts of both analysts. 4 The paper provdes a complete analyss of the two analysts case, and dscusses a possble extenson to a mult analysts setup. Whle the lterature on the ncentves of forecasters, and n partcular analysts, s qute extensve, very lttle has been sad about the order and tmng of analysts forecasts. To the best of my knowledge, the only related theoretcal paper that addresses the endogenous tmng of forecasters s Gul and Lundholm (995). They present a model of two agents, where each has to choose the tmng of hs predcton about the future value of a project. Each agent observes a prvate sgnal of the project s value. They assume that the value of the project equals the sum of the two prvate sgnals. All else equal, the agents prefer to predct sooner rather than later. Contrary to my model, Gul and Lundholm (995) show that agents forecasts always cluster n tme. The setup of my paper s more representatve of analysts envronment and uses less restrctve assumptons. The man d erences n the setup are that, contrary to Gul and Lundholm (995): I do not assume that the sum of the prvate sgnals of the analysts equals the earnngs of the rm; I assume arrval of exogenous nformaton over tme; I assume that the bene t from the forecast depends on the precson of the nvestors belefs, and I explctly model the possblty that forecasts may be based. Several emprcal papers are of partcular relevance: Cooper, Day and Lews (00) nd that lead analysts, dent ed by ther measure of forecast tmelness, have a greater mpact on stock prces than follower analysts. Further, they nd that performance rankngs based on forecast tmelness are more nformatve than rankngs based on abnormal tradng volume and forecast accuracy. Ln, McNchols, and O Bren (003) provde evdence that analysts a laton n uence ther tmelness n downgradng ther recommendatons. Bernhardt and Campello (004) study the relaton between the forecast and ts tmng, but ther focus s mostly whether rms manage analysts forecasts (expectaton management) and on the forecast revsons towards the end of the forecastng perod. 4 The model can be appled to analysts target prces and earnngs growth estmates n a straghtforward way. I conjecture that a smlar argument can be made about stock recommendatons; however, snce those are on a dscrete grd, a d erent methodology must be employed. 3

6 There s an extensve lterature clamng that analysts may have ncentves to bas ther forecasts and recommendatons. 5 Dugar and Nathan (995) show that nancal analysts of brokerage rms that provde nvestment bankng servces to a company are optmstc, relatve to other analysts, n ther earnngs forecasts and stock recommendatons. Ln and McNchols (998) nd that lead and co-underwrter analysts growth forecasts and recommendatons are sgn cantly more favorable than those made by una lated analysts, although ther earnngs forecasts are not generally greater. Mchaely and Womack (999) document that analysts may be too optmstc about rms from whch they are tryng to solct underwrtng busness. Jackson (005) shows that overoptmstc forecasts can be due to trade generaton ncentves. Hong and Kubk (003) show that career concerns may nduce overoptmstc forecasts. Bernhardt and Campello (004) attrbute the bas n the analysts forecasts to expectaton management by the managers, who try to avod negatve earnngs surprses. 6 Lm (00) clams that an analyst s forecast bas s fully ratonal because t nduces the rm s management to produce better nformaton to optmstc analysts. 7 There s an extensve lterature (both theoretcal and emprcal) that examnes the relatonshp between reputatonal concerns and herdng behavor (e.g., Scharfsten and Sten (990), Trueman (994), and Welch (000)). In these models, the reputaton arses from learnng over tme about agent s exogenous characterstcs (e.g. ablty) through hs observed behavor. Consderatons for reputaton or career concerns can lead agents to underweght (or even gnore) prvate nformaton, and to herd. I use a reduced form for the analyst s objectve functon, where forecastng errors nduce reputaton costs. The orgn of the reputaton costs s not modeled drectly; rather t s assumed to be gven exogenously. The rest of the paper s organzed as follows: Secton presents the setup of the model. Secton 3 derves the optmal analyst s behavor n a sngle analyst case. Secton 4 presents and dscusses the tmng game between two analysts. Secton 5 concludes. 5 A very strong mplct testmony, s the Global Settlement (Aprl 003) n whch the largest nvestment banks agreed to pay $.4 bllon n nes and reparatons for potentally msleadng nvestors n ther analysts reports. For detals about the settlement see 6 For earnngs surprse and earnngs management see: Abarbanel and Lehavy (00), Bartov, Gvoly and Hayn (00). 7 Irvne (003) asserts that an analyst s coverage of a rm nduces hgher commssons to hs brokerage rm; nevertheless, analysts can not nduce extra commssons by smply basng ther publshed forecast. 4

7 Model Setup Most of the lterature mplctly assumes that the tmng of analysts earnngs forecasts s random or exogenously determned. The man objectve of analysts s assumed to be the accuracy of ther forecasts,.e., to mnmze the expected squared error of ther forecasts. 8 However, as dscussed n the ntroducton, analysts may have other ncentves that may bas ther forecasts. The ncentves of analysts to bas ther forecasts at a spec c tme (e.g. quarter) are not transparent and not perfectly known to nvestors. The model assumes that the actual bas n analysts forecast may be unknown to the nvestors and to the other analysts (ths s smlar to the assumpton of Fscher and Verreccha (000) about managers reportng bas). A known bas s a partcular case (whch s equvalent to an unbased analysts case). For the smplcty of the dsposton, I assume that the analyst s expected utlty s lnear n hs bas (the lnear coe cent of analyst s denoted by ). Nevertheless, the model s robust to a large class of functonal dependence between the analyst s bas and hs expected payo, and s not restrcted to lnear functonal dependence (later the paper wll elaborate on ths and de ne the larger class of functonal dependence). The above two components of the analyst s utlty functon (precson/reputaton and bas ncentves) are prevalent n the lterature. In a conventonal model based solely on these two components, all analysts would optmally forecast mmedately before the earnngs announcement of the rm, snce ths s when ther forecasts are the most accurate. Ths s clearly not a reasonable outcome, because nvestors would be wllng to reward a devatng analyst who provdes early forecast. So, there must be an o settng e ect that s gnored. The addtonal assumpton that I propose n order to capture ths o settng e ect s the followng: the payo of an analyst depends on the precson of the nvestors belefs about the rm s earnngs mmedately pror to the analyst s forecast. The less precse the nvestors belefs about the rm s earnngs are, the more valuable s the forecast of the analyst to the nvestors, and hence the hgher s the payo of the analyst (for a gven forecast). Followng s the motvaton for ths new assumpton. 8 Whle mnmzng the squared error s the prevalent objectve functon n the lterature, Basu and Markov (003) argue that analysts behavor s ratonal f we assume that they mnmze ther absolute forecast error rather than a quadratc cost functon. 5

8 The analyst s pad by the brokerage house he works for. A bg part of the earnngs of a brokerage house s from tradng commssons from ts nvestors clents. The brokerage house and the analyst want to mantan exstng clents, to attract new clents and to ncrease the volume of trade executed through the brokerage house. The bene t that nvestors receve from analysts earnngs forecasts s early access to nformaton. The most preferred clents of an analyst get hs forecast rst; only later do the less preferred clents get ths forecast, and eventually t s publcly publshed. Access to the nformaton before t becomes publc s so valuable to nvestors that they are wllng to pay for t. The nvestors use ths nformaton n order to form belefs upon whch they make ther nancal decsons. These decsons eventually generate trade n the stock. In the extreme case, where the nvestors are perfectly nformed about the future earnngs of the rm, an analyst s forecast s worthless to nvestors. Moreover, n ths case, an analyst s forecast wll not generate any trade n the stock. The less nformed nvestors are that s, the lower the precson of ther belefs about the rm s earnngs, the more valuable the analyst s forecast s to nvestors and the hgher the trade t may generate. Jackson (005) nds that the reputaton of an analyst ncreases n the tmelness of hs forecasts, whch provdes addtonal motvaton to the assumpton that the expected payo of an analyst decreases n the tme of hs forecast. In nancal markets, as tme advances, more publc nformaton about the forthcomng earnngs arrves. Ths nformaton arrves from analysts forecasts as well as from many other relevant sources of nformaton (Macro economcs, compettors, conference calls etc.). I refer to all nformaton other than analysts forecasts as exogenous nformaton. I denote the precson of the nvestors belefs about the earnngs of the rm at tme t by f (t). I assume that the arrval of the exogenous nformaton s contnuous, meanng that the precson of the nvestors belefs s contnuously ncreasng n tme (n all tmes except at a tme of a forecast publcaton, where there wll be a dscrete ncrease n the precson of the nvestors belefs). 9 All else equal (ncludng the precson of the forecast), the sooner the analyst provdes hs forecast, the more valuable hs forecast s to hs clents and the more trade t may generate hence hs expected payo s hgher. But there s also a cost for early forecastng. The sooner 9 The model s robust to any process of nformaton arrval ncludng dscontnuous processes. Nevertheless, the contnuty assumpton smpl es the analyss and the dsposton by makng the analysts utlty functon d erentable wth respect to tme. 6

9 the analyst publshes hs forecast, the less accurate s the publc nformaton he uses to generate hs forecast; hence, hs expected forecast error s hgher. The basc trade-o that analysts face n determnng the tmng of ther forecasts s between tmelness and precson. Cooper, Day and Lews (00) pont out (emprcally) the wllngness of lead analysts to trade accuracy for tmelness due to ther desre to maxmze compensaton. The model assumes that an analyst has to publsh hs forecast at some pont durng the forecastng season t [0; T ], e.g., between the earnngs report of the prevous perod (quarter) and the forthcomng earnngs report. After the forecastng season, the rm reports ts realzed earnngs, denoted by. 0 At the begnnng of the forecastng season (t = 0), nvestors are assumed to have normally dstrbuted pror belefs about the earnngs of the rm 0 N( 0 ; 0 ). The precson of the pror belefs s denoted by f (0) = 0. As tme progresses, there s a contnuous stream of exogenous nformaton that ncreases the precson of the nvestors belefs f (t) (whle the belefs reman normally dstrbuted). For all t > t we have f (t ) > f (t ), f(0) > 0 and f (T ) <. Whle decdng about the earnngs forecast, an analyst uses all the avalable publc nformaton as well as hs prvate nformaton about the forthcomng earnngs of the rm. I assume that analyst gets a prvate sgnal about the earnngs of the rm - e = +e", where e" N 0; " s ndependent of (n the case of more than one analyst, for all 6= j e" s ndependent of e" j ). I denote the precson of analyst s prvate sgnal by f S ". There exsts an extensve lterature on how forecasters and n partcular analysts decde about the weght they place on prvate and publc nformaton (e.g. Chen and Jang (005)). I assume that whle formng ther belefs, analysts are ratonal and apply base rule. As to the tme at whch an analyst observes hs prvate sgnal, n the model t does not n uence the equlbrum results (as long as t happens before the equlbrum tmng of hs forecast). For smplcty, let s assume that the analysts get ther prvate sgnals at t = 0. I use a reduced form of the analyst s objectve functon, whch captures the above characterstcs and trade-o. The expected utlty of analyst who makes a forecast at tme t s assumed 0 Assumng that companes manpulate the reported earnngs by a constant (see for example Sten (89) and Guttman, Kadan, Kandel (005)), would not n uence the results. It could be an approxmaton of a dscrete process of normally dstrbuted sgnals. 7

10 to be: EUt = F ;t E [j ; I t ] h E F ;t j ; I t f (t) () where N( ; ) s the forecast bas parameter, s a postve constant, F ;t denotes the forecast of analyst (publshed at tme t), s the prvate sgnal of the analyst, and I t s all the publc nformaton avalable at tme t (mmedately pror to the analyst s forecast) whch ncludes the precedng analysts forecasts. The realzaton of s known only to the analyst hmself, where s common knowledge. E s the expectaton operator. The utlty functon of the analyst has three components: the term F ;t E [j ; I t ] h captures the analyst s ncentves to bas hs forecast; The term E j ; I t s the mean squared error (MSE) of the analyst s forecast and captures the analyst s desre to be precse (reputaton/precson); and f (t) captures the ncentve of the analyst to provde hs forecast at a stage where the precson of the nvestors belefs s low (tmelness). Wth out loss of generalty the coe cent of f (t) s normalzed to be one. I beleve that the three ncentves of analysts mentoned above are the central ones to ther behavor. As n every model, there may be other ncentves of analysts whch are not accounted for n the model. Note that mmedately followng the analyst s forecast, there s a dscrete jump n the precson of the nvestors belefs, and thereafter, untl the forecast of the next analyst, the precson of the nvestors belefs contnuously ncreases accordng to the exogenous nformaton process. Usng the above setup, I rst solve the optmzaton problem of a sngle analyst case hs optmal forecastng tmng and the optmal forecast at that pont n tme. Next, I study a game between two analysts, who must decde at what pont n tme to publsh ther forecasts. Snce the analyst s expected utlty depends on the precson of the publc s nformaton at the tme of A more general utlty functon s: EUt = F ;t E [j ; I t ] h E j ; I t g (f (t)) where g () s a contnuously ncreasng functon. I show n Appendx that the results of the model holds for a very bg set of functons g (f (t)). w.l.o.g. s normalzed to equal. An alternatve way for modelng, s to assume a multplcatve utlty functon rather than an addtve one, for example EUt = E h F t f (t) F t R j ; It. In ths case, the optmal forecastng tmng s typcally a corner soluton. Nevertheless, n general, the comparatve statcs work (weakly) n the same drectons as n the paper s addtve utlty functon. F ;t F ;t 8

11 forecast (and not on the tme per se), a strategc nteracton arses between the analysts, whch should be consdered. I derve a pure strateges Subgame Perfect Equlbrum of ths game and prove ts exstence and unqueness. 3 The Sngle Analyst Case Ths secton analyzes the case of a sngle analyst who has to choose the tme of hs forecast. Ths smple case s not a strategc game, but rather a smple optmzaton problem. The analyst has to decde about the tme of hs forecast and the value to be forecasted. Whle solvng for the optmum I rst derve the optmal forecast for every possble forecastng tme, then, gven the optmal forecasts, I nd the optmal tmng of the forecast (to be precse, I derve the precson of the nvestors belefs at whch t s optmal for the analyst to publsh hs forecast). 3. The Optmal Forecast The analyst s Frst Order Condton wth respect to the forecast at a gven tme t s E F ;t j ; I t = 0: The Second Order Condton for maxmum s sats ed. Hence, for every gven tmng of forecast t, the analyst s optmal forecast s: F ;t = + E (j ; I t ) () = + " + t + " + " where t = E (ji t ) and = V ar (ji t ) are the mean and varance of nvestors belefs at tme t mmedately pror to the forecast respectvely (Note that t may be d erent from 0 ). The optmal forecast of the analyst s to bas hs forecast by the constant. Although the analyst s optmal forecast s lnear n hs prvate sgnal, t does not fully reveal hs prvate sgnal snce s not known to nvestors. Only n the case where the bas parameter s common knowledge, the analyst s forecast fully reveals hs prvate sgnal. 9

12 Substtutng the optmal forecast of the analyst nto hs utlty functon yelds: EU t = 4 f (t) + f S f (t) : 3 (3) An ncrease n f (t) has two opposte e ects on the analyst s expected utlty. On the one hand, t ncreases the analyst s nformaton and reduces the expected cost of a forecast error (reduces f(t)+f S ). On the other hand, the analyst ncurs the drect cost of a later forecast. In the next secton, I nd the analyst s optmal behavor n solvng the above trade-o. 3. The Optmal Tmng of the Forecast The tme per se s not an mportant factor, rather, what matters s the precson of the nvestors belefs at each pont n tme. 4 In lght of the above trade-o, n order to nd the precson of the nvestors belefs at the optmal forecastng tmng, I take the dervatve of the analyst s expected utlty (equaton 3) wth respect to the precson of the nvestors belefs. Optmzaton yelds (f (t) + f S ) = 0; whch suggests that the analyst should forecast at f (t) = p f S : But, the precson of the nvestors belefs at whch the analyst can forecast s constraned by the precson of the nvestors belefs at the begnnng and at the end of the forecastng season,.e. f (0) and f (T ). I denote the analyst s optmal forecastng tme by t. Incorporatng the constrant of the forecastng season, we get: Proposton 8 >< t = 0 f f S p f (0) f p f S f p f (T ) < f S < p f (0) >: T f f S p >; f (T ) 3 EUt = E E [j ; I t ] + j ; I t f (t) = 4 V ar(j ; I t ) f (t). n the appendx I show that V ar(j ; I t ) = f(t)+f S 4 The tme value of money could be accounted for as well, wthout qualtatvely a ectng the results. 9 >= 0

13 where f p f S s the tme at whch the precson of nvestors belefs s f (t) = p f S. 5 Hereafter, I refer to the tme t as the unconstraned optmum tmng. If the analyst s prvate sgnal s su cently precse, he wll publsh hs forecast mmedately at the begnnng of the forecastng perod, snce an ncrease n the precson of the publc nformaton (and hence n hs forecast) s not su cent to compensate for the assocated cost of late forecast. Ths means that hs expected utlty monotoncally decreases n tme, and hence he forecasts as early as he can. Ths s llustrated by the Hgh Precson case n the gure below. On the other hand, for su cently low precson of the analyst s prvate sgnal, the analyst wll wat to gan as much publc nformaton as he can, and wll forecast at T: See the Low Precson case n the gure below. In the ntermedate case, the trade-o s such that the analyst wats untl tme t where f (t ) = p f S. After ths tme, the cost of late forecastng exceeds hs payo from the ncrease n the precson of the publc nformaton he can use. Ths mples that for p f (T ) < f S < p f (0): the expected utlty of the analyst monotoncally ncreases for f (t) < f (t ), and after the tme t t monotoncally decreases. Ths s llustrated by the Interor soluton case n the gure below.. EU t Low precson f(0) Interor soluton Hgh precson f(t) Expected utlty - d erent cases f(t) 5 If we assume that the coe cent of the bas n the analyst s expected payo, s not a constant and s a functon r of f(t) (.e. (f(t)) and not just ), then the optmal forecast tme of the analyst s: f (t ) = f S. The model s robust to all (f(t)) for whch the above f (t ) s well de ned.

14 Corollary (Comparatve Statcs) Gven the nteror soluton for t,.e. p f (T ) < f S < p f (0), the optmal forecastng tme of the analyst decreases n the precson of hs prvate sgnal (f S ); and ncreases n hs error/reputaton cost parameter ( ). The above corollary s qute ntutve. Less precse prvate sgnal of the analyst nduces hm to postpone hs forecast and gan from the ncreased precson of the publc s nformaton. On the other hand, the lower hs reputaton cost for a gven forecast error (captured by ), the hgher hs propensty to rsk a large error n order to provde hs forecast at an early stage. Snce the coe cent of the precson of the nvestors belefs n the analyst s utlty functon s normalzed to equal, hgher rewards for early forecast s equvalent to reducng both and, and nduces an earler forecast. The tmng of the analyst s forecast s ndependent of the precson of the nvestors belefs about hs bas. The realzatons of both the prvate sgnal and the bas parameter do not a ect the optmal forecastng tmng. The realzed sgnal does not n uence the precson of the analyst s belefs nor the precson of the nvestors belefs followng hs forecast (t n uences only the condtonal expectatons but not the condtonal varance). Hence, the realzed sgnal does not n uence the trade-o that the analyst faces whle choosng the tme of hs forecast. As to the value of, changes n lnearly change the bas n the analyst s forecast. All else equal, a change n a ects only the rst expresson n (3) whch s ndependent of f (t), and s outsde the analyst s tmng trade-o decson. From the nvestors perspectve, the closer the realzed value of to ts mean, the more accurate ther belefs followng the analyst s forecast are. The more con dent the nvestors are regardng the analyst s bas, the more they learn on average from hs forecast,.e. the hgher the weght they attach to the forecast. In the next secton I ntroduce a tmng game between two analysts. Due to the strategc nteracton between the analysts, both ther pror belefs about the bas of the other analyst and the precson of the prvate sgnals wll n uence ther equlbrum strateges. 4 Tmng Game Wth Two Analysts Most stocks are covered by more than one sell-sde analyst. Ths mples that t s mportant to understand how competton alters the behavor of sell-sde analysts. Recall that followng

15 f(t) f Sj 0 t π F j,t Fgure : Two D erent Tme Lnes an analyst s forecast, there s a dscrete ncrease ( jump ) n the precson of nvestors belefs. Hence, the support of the precson of nvestors belefs at whch an analyst can publsh hs forecast s no longer the entre nterval [f(0); f (T )], as t was n the sngle analyst case. In the context of a two-analysts game, one can thnk of two d erent tme lnes: the calendar tme lne and the precson of the nvestors belefs tme lne. Whle the calendar tme lne s contnuous, the precson of the nvestors belefs tme lne has a jump at the tme of an analyst s forecasts. Fgure presents the two tme lnes for analyst. The horzontal axes (the calendar tme lne) obtans contnuous values, but on the vertcal axes (the precson of nvestors belefs at whch analyst can publsh hs forecast) there s a dscrete jump ("hole") followng the forecast of analyst j at tme t 0. Note that the locaton of ths "hole" s determned endogenously, accordng to the equlbrum tmng strateges of the analysts. The sze of the dscrete jump followng the forecast of analyst j s denoted by f Sj. Whle determnng the optmal forecastng tmng, the analyst cares about the precson of the nvestors belefs, and not on the tme per se. The fact that a forecast of an analyst changes the precson of the nvestors belefs at whch the other analyst can publsh hs forecast mmedately ntroduces strategc nteracton between the two analysts. Let s assume, for example, that the unconstraned optmal forecastng tme for analyst s t. If he wats tll t, he bears the rsk that analyst j wll step n front of hm and forecast at t ". If analyst j does forecast at t ", analyst wll face nvestors belefs wth precson of (f (t ")) + f Sj (where f Sj denotes the ncrease n the precson of the nvestors belefs due to the forecast of analyst j). Ths wll decrease the expected utlty of analyst relatve to forecastng rght before analyst j. Analyst j has to take nto account that analyst may hence forecast earler than t, and should 3

16 consder forecastng even earler. Ths example llustrates the knd of strategc nteracton that the analysts have to take nto account. In ths secton, I develop and prove the exstence and unqueness of a Subgame Perfect Equlbrum n pure strateges of the tmng game between two analysts. In the game wth two analysts, the second forecaster ncorporates the nformaton from the rst forecast. The hgher the precson of the nvestors (and the other analyst s) belefs regardng the bas of an analyst, the more the nvestors can nfer from hs forecast, and the hgher s the precson of ther belefs mmedately after hs forecast. Hence, the magntude of the dscrete jump n the precson of the nvestors belefs followng an analyst s forecast ncreases n both the precson of the analyst s prvate sgnal and n the precson of the nvestors belefs about hs bas. Whle formng hs strategy, an analyst has to take nto account the ncrease n the precson of nvestors belefs due to the other analyst s forecast, and due to hs own forecast. In contrast to the sngle analyst case, the precson of the nvestors belefs regardng the bas of the analysts does n uence ther equlbrum tmng. For smplcty, I rst solve the model for the partcular case where the bas of each analyst s common knowledge, and hence an analyst s forecast fully reveals hs prvate sgnal. 6 After ths basc model s establshed, the case of asymmetrc nformaton regardng the analyst s bas s ntroduced. The basc setup and assumptons are smlar to the sngle analyst case. 4. The Known Bas Case Let s assume that there are two analysts = ;. The expected utlty of analyst who makes a forecast at a tme t (at whch the precson of the nvestors belefs s f (t)) s assumed to be (as before): EUt = F ;t E [j ; I t ] h E F ;t j ; I t f (t) ; where the parameters of the utlty functon (ncludng ) and the precson of the prvate sgnals of the analysts are common knowledge. 6 As wll be shown ahead, the equlbrum forecast s lnear n the prvate sgnal, and hence the forecast fully reveals the prvate sgnal. 4

17 Followng the arrval of exogenous publc nformaton, the precson of the nvestors belefs s contnuously ncreasng n tme, except at the tmes of the analysts forecasts, where t follows a dscrete jump. At the begnnng of the game (or at any tme before the analyst s forecast), each analyst observes a prvate sgnal of the rm s earnngs e = + e". Each analyst has to forecast at some pont durng the forecastng season t [0; T ]. An assumpton regardng the possblty that both analysts forecast smultaneously at a gven pont of tme s n place. I assume that analysts can forecast smultaneously at t = 0 or at t = T, n whch case the utlty of each of them s exactly the same as f he were the only analyst to forecast at that pont (smlar to the sngle analyst case). For smplcty of dsposton, I assume that for t (0; T ) there can be two consecutve forecasts at the same nstant of tme, so there s no smultaneous forecastng. Ths assumpton holds by applyng the framework of Smon and Stnchcombe (989) (see the relevant footnote n Proposton ). 7 A strategy for an analyst s a functon that maps from the pror parameters and the analyst s nformaton nto a precson of nvestors belefs at whch to forecast, and the forecast tself at that tme. The pror parameters nclude: the utlty functon of each analyst, the precson of the analysts prvate sgnals, and the precson of the nvestors belefs about the bas parameters of the analysts. I denote the precson of the nvestors belefs at the equlbrum forecastng tme of analyst by f t;c (where the subscrpt c ndcates the constrant on the precson of the nvestors belefs at whch an analyst can forecast due to the dscrete jump n the precson of the nvestors belefs followng the other analyst s forecast). If the ncrease n the precson of nvestors belefs followng the analysts forecasts s su - cently small, and the unconstraned optmal forecastng tmes of the analysts are su cently apart from each other, then t s feasble that each analyst forecasts at hs unconstraned optmal forecastng tme t (the tmng of the sngle analyst case). In ths case, none of the analysts has an ncentve to devate from ths strategy, hence t s an equlbrum. Moreover, I later show that n ths case t s the unque Subgame Perfect Equlbrum n pure strateges. But f t s not feasble, then each analyst has to take nto account the ncrease n the precson 7 Ths assumpton wll be needed to show the unqueness of the Subgame Perfect Equlbrum. In the case that there s some uncertanty regardng the parameters or regardng the equlbrum tmng, or, f we use a stronger concept of equlbrum lke " equlbrum (or tremblng hand perfecton) the unqueness holds even wthout the above assumpton. 5

18 of nvestors belefs due to hs own forecast and the other analyst s forecast. The followng Lemma descrbes the ncrease n the precson of nvestors belefs due to an analyst s forecast. Lemma When s common knowledge, the ncrease n the precson of the nvestors belefs due to an analyst s forecast s constant (.e., ndependent of the precson of the nvestors belefs at the tme of forecast), and s equal to the precson of the analyst s prvate sgnal. For proof see Appendx.A. If the precson of nvestors belefs mmedately before the analyst s forecast s f (t), and the precson of the prvate sgnal of analyst s f S ", then the precson of the nvestors belefs mmedately after the forecast of analyst s f (t) + f S. I next derve the pure strateges Subgame Perfect Equlbrum. I start by dervng the condtons for the corner solutons, then I ntroduce a de nton that wll be used n Proposton () that provdes the full spec caton of the equlbrum. Clam For each analyst = ; : (A) If f S > p f(0) then he forecasts as soon as he can; that s at t = 0. (B) If f S < p f (T ) then the analyst forecasts at the latest possble tme; that s at t = T. Proof. Ths s the unconstraned optmal strategy of an analyst, and t s feasble for both analysts. By revealed preferences ths s the optmal strategy. Let s consder the case where the unconstraned optmum of both analysts s nteror, that s for = ; f(0) < f (t ) < f (T ). Gven that the ncrease n the precson of nvestors belefs due to the forecast of analyst s f S, there s a hole of sze f S n the support of the precson of nvestors belefs at whch analyst can publsh hs forecast. But the locaton of ths hole depends on the tmng strategy of analyst, whch of course, takes nto account the strategy of analyst. To resolve ths strategc nteracton I wll de ne and use the "nd erence nterval". Intutvely speakng, the nd erence nterval of analyst s the nterval of precson of nvestors belefs, of sze f S, for whch analyst s nd erent between forecastng at ether the lower or the upper end of ths nterval. Note that n the nteror unconstraned optmum case, the expected utlty of an analyst monotoncally ncreases n the precson of the nvestors belefs untl t gets maxmzed, and from there on the expected utlty monotoncally decreases. Ths mples 6

19 both the unqueness of the nd erence nterval and that the nd erence nterval straddles the unconstraned optmum of the analyst (where ts expected utlty s maxmzed). Bellow s a formal de nton of the lower end of the nd erence nterval (fl ), whch also de nes the nd erence nterval for more complex cases where the above "ntutvely speakng de nton" does not exst. De nton De ne f L as follows: If there exsts a precson of nvestors belefs f 0 such that analyst s nd erent between forecastng at a precson of nvestors belefs that equals ether f 0 or that equals f 0 + f S f L f 0. 8 Fgure. llustrates f L for ths case (nteror nd erence nterval). If an nd erence nterval as the above does not exst, then t s one of the followng two cases: Case A - EU (f (t) = f (0)) > EU (f (t) = f (0) + f S ). 9 In ths case I de ne f L = f (0). (See Fgure.) Case B - EU (f (t) = f (T )) > EU (f (t) = f (T ) f S ). 0 In ths case I de ne f L then to be the precson of nvestors belefs where EU (fl ) = EU (f (T )) and fl < f (T ). (See Fgure.3) EU Indfference Interval F L f(t)* f L +f S f(t) Fgure.: nteror nd erence nterval 8 If such f 0 exsts then t s unque and f (0) fl < f (t ). 9 Where EU (f (t) = f (0)) s the expected utlty of analyst gven that he forecasts at the tme where f (t) = f (0). 0 Up to ths pont, for the smplcty of dsposton, I have not de ned whether f (T ) s the precson of nvestors belefs at the end of the forecastng season gven that the other analyst has or has not publshed hs forecast. Here I can no longer be vague about t, and I de ne f (T ) as the precson of nvestors belefs gven that the other analyst has publshed hs forecast. 7

20 EU EU Indfference Interval Indfference Interval f L = f(0) f(0)+f S Fgure.: Case A f(t) f(t)-f S f L f(t) f(t) Fgure.3: Case B Intutvely, f L s the lower end of the nterval of precson of nvestors belefs of sze f S, that wll make the analyst nd erent whether he forecasts at the one or the other end of that nterval. Gven the above de nton, I can now present the man proposton of ths secton. Proposton There exsts a unque Subgame Perfect Equlbrum n pure strateges. equlbrum strateges of the analysts are as follows: For each analyst = ; f fl = f (0) he forecasts at t = 0. If for at least one of the analysts f L > f (0), then analyst s the rst to forecast f and only f f L < f L. The Let s assume w.l.o.g. that analyst s the rst to forecast. Analyst forecasts at a tme t ;c where the precson of the nvestors belefs s: f t ;c = Mn f L ; f (t ) : If analyst forecasts at fl then the analyst forecasts mmedately after, where the precson of the nvestors belefs s f (t) = f L + f S. 3 If the rst analyst forecasts at f (t ) then the second analyst wll forecast at f (t ) f t s feasble (that s f: f (t ) + f S f (t )), or else In Case B the nterval s smaller than f S. I could use a d erent te breakng rule where analyst s the rst to forecast fl f L. 3 The strategy whch predcts that the second analyst wll forecast mmedately after the rst analyst s somewhat vague. One way to have well de ned strateges and outcomes s usng the framework of Smon and Stnchcombe (989). All three assumptons that they mpose on the strateges (F-F3) hold n my model. Usng ths framework, there can be two consecutve forecasts at the same nstant of tme (they show that the lmt of the dscrete tme equlbra as the tme nterval goes to zero converges to the contnuous tme equlbrum). Another way to have well de ned strateges s usng the framework of Perry and Reny (994), where restrcton (S4) upon strateges mposes some " lag between agents actons and guarantees that the game s well de ned. Adoptng the framework of Perry and Reny requres some adjustments for fl (n a magntude smaller than "). 8

21 mmedately after the rst forecast. The optmal forecast of every analyst s: F ;t = + E (j ; I t ) : The o equlbrum belefs are as follows. Let s assume w.l.o.g. that f L < f L. For all f (t) > f L, f no forecast was publshed analyst beleves that analyst s gong to forecast mmedately. 4 Before provng the proposton, I descrbe the equlbrum ntutvely and graphcally. The equlbrum may have one of the followng two patterns. Non Clusterng Equlbrum Pattern (separaton n tme) each analyst publshes hs forecast at hs unconstraned optmum. 5 If ths s not feasble, then the equlbrum s of the second pattern. Clusterng n Tme Pattern the rst forecaster s the analyst who s lower end of the nd erence nterval (f L ) s smaller. Assume ths s analyst. Analyst publshes hs forecast at fl. Followng hs forecast, the precson of the nvestors belefs ncreases nstantaneously and becomes hgher than the unconstraned optmum of analyst. At that pont, the expected utlty of analyst decreases n the precson of the nvestors belefs, and hence he publshes hs forecast mmedately. Both patterns are presented n the followng two gures. EU Non-Clusterng Pattern U (f(t)) U (f(t)) f(0) f(t *),c =f(t *) f(t,c *)+f S f(t *),c =f(t *) f(t) Analyst forecasts frst at hs unconstraned optmum - f(t *). Followng hs forecast the precson of the nvestors belefs jumps to f(t,c *)+f S, whch s stll lower than the unconstraned optmum of analyst. Analyst wats untl the precson of the nvestors belefs equals hs unconstraned optmum and then publshes hs forecast. 4 The equlbrum can be supported by a larger and more general set of o equlbrum belefs, ncludng mxed strateges o the equlbrum path. 5 The Non-Clusterng Pattern may be feasble even n the case where the unconstraned optmum of an analyst s ncluded n the nd erence nterval of the other analyst. 9

22 EU Clusterng Pattern f(0) f L U (f(t)) f f L+f S f L L+f S f(t *),c f(t *),c U (f(t)) Snce f L<f L analyst s the frst to forecast. He publshes hs forecast at f L. Followng hs forecast, the precson of the nvestors belefs jumps to f L+f S whch s hgher than the unconstraned optmum of analyst. Snce the expected utlty of analyst at ths regon s decreasng n the precson of the nvestors belefs, analyst publshes hs forecast mmedately after the forecast of analyst. Proof of Proposton. Analyst wll never forecast earler than f L f(t) snce he can guarantee hmself an expected utlty of at least EU (f (t) = f L ) = EU (f (t) = f L + f S ). If the precson of nvestors belefs s hgher than f (t ) (due to a dscrete jump after the forecast of analyst ), then analyst forecasts mmedately. Hence, the only nterval of precson of nvestors belefs left to nvestgate (from analyst 0 s perspectve) s f L < f (t) < f (t ). In the case where fl > f (t ) analyst wll not forecast before fl, and analyst wll wat and forecast at hs unconstraned optmum (where the precson s f (t )). In the case where f L < f L < f (t ) t s straghtforward that analyst wll wat at least untl f L. Before proceedng wth the proof of the Proposton, I ntroduce the followng Lemma. Lemma If f L < f L < f (t ) then there s no pure strateges Subgame Perfect Equlbrum n whch the rst forecast of the analysts wll be at f (t) > f L. Proof of the Lemma. Assume that such pure strateges Subgame Perfect Equlbrum exsts. Then, the second forecaster can devate and forecast at a su cently small amount of tme earler than the rst forecaster, and by dong so he strctly ncreases hs expected utlty n contradcton to ths beng an equlbrum. QED Lemma. The Lemma ndcates that for fl < f L < f (t ) ; f analyst has not forecasted before fl, analyst wll forecast at fl. 0

23 So far, I have shown that f fl < f L then analyst wll forecast rst at a precson of nvestors belefs of Mn ffl ; f (t )g. It s straghtforward that the second analyst to forecast wll ether wat untl f (t ) (f t s feasble, that s f: f (t ) > fl + f S ), or else he wll forecast mmedately after the rst analyst. QED. 4.. Dscusson of the Known Bas Case The central questons of ths paper are: what determnes the order and tmng of the analysts forecasts, and what are the emprcal predctons generated by the model? In the model, the order and tmng of the forecasts are determned by the followng factors: the precson of the prvate sgnals of the analysts (f S ), the reputaton parameters of the analysts ( ), and the process of exogenous nformaton arrval (n the Unknown Bas case that wll be presented n the followng secton, also the precson of the nvestors belefs about the bas parameter, n uences the order and tmng of the forecasts). I next elaborate on the n uence of each of the above factors on the equlbrum order and tmng of the analysts earnngs forecasts. An ncrease n the reputaton cost parameter of analyst ( ) motvates hm to publsh hs forecast later when there s more publc nformaton that he can use. More formally, t pushes the analyst s unconstraned optmum to later n tme where the precson of the nvestors belefs s hgher. Note that the sze of the nd erence nterval of both analysts s ndependent of. Hence, an ncrease n shfts the nd erence nterval of analyst to the rght, wthout a ectng the nd erence nterval of the second analyst. Ths wll nduce a later forecast by analyst (except for the clusterng pattern case where analyst forecasts mmedately after the other analyst, and stll does so after the ncrease n ). Note that f analyst was the rst to forecast, the ncrease n may also change the order of the forecasts. Changes n the process of exogenous nformaton arrval wll not n uence the precson of the nvestors belefs at whch each of the analysts wll publsh hs forecast. That s, wth respect to the "precson of the nvestors belefs tme lne" nothng changes. The only thng that does change s the calendar tme at whch each analyst publshes hs forecast. An ncrease n the precson of the prvate sgnal of analyst has two con ctng e ects on the equlbrum order of forecasts. On the one hand, the unconstraned optmum of analyst s now at a lower precson of nvestors belefs. On the other hand, the dscrete jump n the precson

24 of nvestors belefs followng the forecast of analyst (f S = f S ) becomes bgger, whch ncreases the nd erence nterval of analyst j, and reduces the lower end of the nd erence nterval of analyst j f j L. Ths n turn (all else equal), nduces earler forecast by analyst j (n the clusterng pattern equlbrum). The corollary bellow ndcates that the n uence of the rst e ect on the order of forecasts always domnates. Corollary (Comparatve Statcs) Suppose that n equlbrum, analyst forecasts at tme t 0 (0; T ). An ncrease n the precson of the prvate sgnal of analyst wll advance the tme of hs forecast, and weakly advance the forecastng tme of analyst j. If analyst was the rst forecaster, he wll stll forecast rst, but f he was the second forecaster, then he may now become the rst to forecast. Moreover, the precson of the nvestors belefs mmedately pror to the rst forecast wll be lower relatve to the tme before the change. 6 For the proof of the Corollary see Appendx 3. An nterestng feature of the equlbrum s that t s not necessarly the analyst wth the hgher precson of prvate sgnal who wll be the rst to forecast. The unconstraned optmum of an analyst s determned by the combnaton of the precson of hs prvate sgnal (f S ) and the error/reputaton cost parameter ( ). It s possble that analyst has a hgher precson of prvate sgnal, but hs error cost parameter s su cently hgher than that of analyst j, so that the unconstraned optmum of analyst wll be at a hgher precson of the nvestors belefs. Ths ndcates that when comparng two d erent sngle analyst cases, t mght be that the unconstraned optmum of an analyst wth hgher precson of prvate sgnal s at a later tme than of the other analyst. But n the forecastng tmng game, t s also possble that even though the unconstraned optmum of analyst s at a lower precson of nvestors belefs than the unconstraned optmum of analyst j, stll analyst j wll be the rst to forecast. Ths can happen because the hgher precson of the prvate sgnal of analyst nduces a bgger nd erence nterval of analyst j, whch causes f j L to be lower than f L. Snce the rst analyst to forecast s the one who demonstrates a smaller lower end of the nd erence nterval (fl k ), t mght be that f j L < f L. If ths s the case, then analyst j wll be the rst to forecast. 6 The precson of nvestors belefs at the tme of the second forecaster may be hgher, the same or lower. Both cases are presented n the proof.