Dynamic Forecasting Behavior by Analysts: Theory and Evidence

Transcription

1 Dynam Foreastng Behavor y Analysts: Theory and Evdene Jonathan Clarke Ajay Sramanan DPree College of Management Georga Insttte of Tehnology 755, Ferst Drve, Atlanta, GA 3033 Marh 003

2 Dynam Foreastng Behavor y Analysts: Theory and Evdene Astrat We examne the dynam foreastng ehavor of nvestment analysts n resonse to ther ror erformane relatve to ther eers wthn a ontnos tme/mlt-erod framework. Or model redts a U-shaed relatonsh etween the oldness of an analyst's foreast, that s, the devaton of her foreast from the onsenss and her ror relatve erformane. In other words, analysts who sgnfantly ot erform or nder erform ther eers sse older foreasts than ntermedate erformers. We then test these redtons of or model on oserved analyst foreast data. Consstent wth or theoretal redtons, we doment an aroxmately U-shaed relatonsh etween analysts' ror relatve erformane and the devaton of ther foreasts from the onsenss. Or theory examnes the mat of oth exlt nentves n the form of omensaton strtres and mlt nentves n the form of areer onerns, on the dynam foreastng ehavor of analysts. Consstent wth exstng emral evdene, or reslts mly that analysts who fae greater emloyment rsk that s, the rsk of eng fred for oor erformane have greater nentves to herd, that s, sse foreasts that devate less from the onsenss. Or mlt-erod model allows s to examne the dynam foreastng ehavor of analysts n ontrast wth the extant two-erod models that are stat n natre. Moreover, the model also dffers sgnfantly from exstng theoretal models n that t does not rely on any sef assmtons regardng the exstene of asymmetr nformaton and/or dfferental analyst altes. Key Words: Analysts, Career Conerns, Dynam Foreastng, Herdng

3 . Introdton We examne the dynam foreastng ehavor of nvestment analysts n resonse to ther ror erformane relatve to ther eers wthn a ontnos tme/mlt-erod framework. Or model redts a U-shaed relatonsh etween the oldness of an analyst's foreast, that s, the devaton of her foreast from the onsenss, and her ror relatve erformane. In other words, analysts who sgnfantly ot-erform or nder-erform ther eers sse older foreasts than ntermedate erformers. We then test these redtons of or model on oserved analyst foreast data. Consstent wth or theoretal redtons, we doment an aroxmately U-shaed relatonsh etween devaton of analysts foreasts from the onsenss and ther ror relatve erformane. Or model reles on two assmtons: an analyst's omensaton s onvex n her erformane relatve to her eers and faes sgnfant negatve areer onerns, that s, the rsk of losng her jo for oor relatve erformane. These assmtons are onsstent wth exstng emral and anedotal evdene. Althogh data on analysts omensaton s not avalale, Wse 000 notes that there are large ay dsreanes etween analysts eng named to Instttonal Investor s All-Amera researh team and those that do not make the team. Gven that eone and W 00 fnd that Instttonal Investor all-stars have seror erformane to non-star analysts, ths wold mly onvexty n the omensaton strtres for analysts. Mkhal, Walther, and Wlls 999 fnd that an analyst s more lkely to e fred f her foreast aray delnes relatve to her eers. However, they fnd no assoaton etween an analyst s roalty of trnover and asolte foreast error. In other words, ths sggests that t s relatve, rather than asolte erformane, that matters. The model we roose and nvestgate an e refly desred as follows. At eah foreastng date, an analyst faes the hoe etween a old strategy and a onservatve strategy.

4 The analyst's foreast devates to a greater extent from the onsenss foreast nder the old strategy than the onservatve strategy. The old and onservatve strateges may reslt from rvate sgnals reeved y the analyst or from lly avalale nformaton aot the frm eng overed. Under ether strategy, the analyst has a nonzero roalty of ether oterformng or nder erformng the medan analyst. The analyst s omensated at the end of eah foreastng erod and her omensaton s a onvex fnton of her ror relatve erformane over some tme horzon. There s a nonzero roalty that the analyst may e fred f her relatve erformane s elow an exogenos level. The analyst has nter-temoral referenes for the erod ash flows that omrse her omensaton and dynamally hooses the old or onservatve strategy at eah date to maxmze her exeted tlty. We assme that the analyst has lnear referenes rely for smlty. Or analyss an easly e generalzed to nororate rsk averson. We exltly solve the analyst's stohast dynam otmzaton rolem and show that the otmal oly for the analyst an e desred as follows: there exst two thresholds of ror relatve erformane sh that f the analyst ether oterforms the hgher threshold or nder erforms the lower threshold, she hooses the old strategy and f her ror relatve erformane les etween the thresholds, she hooses the onservatve strategy. Therefore, the analyst makes a old foreast f she ether sgnfantly ot-erforms or nder-erforms the medan analyst and makes a onservatve foreast f she s an ntermedate erformer. The ntton for these reslts s the followng. When the analyst sgnfantly oterforms the medan analyst, the onvexty of her omensaton strtre ndes her to take on the nreased rsk of ssng a old foreast. On the other hand, f she sgnfantly nder erforms the medan, she faes a sstantal rsk of eng fred for oor relatve erformane. Therefore, she takes on the nreased rsk of the old strategy to nrease the roalty that her erformane wll rse aove

5 the level where she may e fred. At ntermedate levels of ror relatve erformane, the analyst trades off the hgher exeted omensaton from hoosng the old strategy for the lower roalty that her erformane may delne elow the level where she may e fred from hoosng the onservatve strategy. In general, there exsts a nonemty ntermedate regon of ror relatve erformane where she refers the onservatve strategy. We also show that f the analyst faes lttle or no rsk of eng fred, she wll always hoose the old strategy. Ths reslt follows easly from the onvexty of her omensaton strtre. Or reslts have mmedate mlatons for herdng y analysts. In artlar, they show that ntermedate erformers tend to devate less from the onsenss, that s, herd more than sgnfant ot-erformers and nder-erformers. Moreover, or reslt that an analyst always hooses the old strategy f she faes lttle or no emloyment rsk ndates that the nentve to herd nreases wth emloyment rsk. In artlar, ths mles that more exerened analysts who fae lower emloyment rsk herd less than less exerened analysts. Ths redton s onsstent wth the emral reslts of Hong, Kk, and Solomon 000 who examne a samle of stok analysts' earnngs foreasts and show that yonger analysts tend to herd more than older, more exerened analysts who fae lower emloyment rsk. Or reslts therefore hghlght the mat of oth exlt nentves n the form of omensaton strtres and mlt nentves n the form of areer onerns on the foreastng ehavor of nvestment analysts. Several reent aers have emhaszed the mortane of areer onerns n varos ontexts. Sharfsten and Sten 990, Prendergast and Stole 996 and Morrs 997 arge that areer onerns may nde ororate and/or fnd managers to gnore rvate nformaton and follow the herd or avod followng t when ther atons are oservale. In Sharfsten and Sten 990, "smart" managers reeve orrelated nformaton, whle "dm" Chevaler and Ellson 999 otan smlar reslts n ther emral nvestgaton of mtal fnd managers.

6 managers reeve norrelated nose. Ths, a manager who learns that hs rvate nformaton dffers from that of another manager eleves that he s more lkely to e "dm", gnores hs nformaton and "herds". In Prendergast and Stole 996, managers have rvate nformaton aot the reson of ther nformaton. A older aton sgnals that a yong manager knows hs nformaton to e good, and hene yong managers have an nentve to take exessvely old atons. Older managers have an nentve to eome jaded and do not hange ther atons a great deal from erod to erod. Zweel 995 rooses a model of the ehavor of ororate managers where takng an noserved nonventonal aton nreases the varane of the market's ex ost assessment of a manager's alty. In a reslt that s remnsent of the reslts that we otan, Zweel 995 shows that average managers refer the onventonal aton ease t redes the rsk of ther eng fred, whle hgh or low alty managers may refer nonventonal atons. Or aer dffers sgnfantly from the aove aers n that we fos on nvestgatng the dynam foreastng ehavor of analysts n a mlt-erod/ontnos tme framework, rather than the two-erod models examned n the aove aers. Welh 000, n artlar, notes that these exstng stat theores are desgned to exlan a steady state n whh all analysts herd erfetly, not to exlan an ever varyng tme-seres of reommendatons or a resdal dfferene n onons aross analysts. g Or dynam model attemts to address some of these onerns. Moreover, n ontrast wth the aove aers, we do not make any sef assmtons aot whether analysts ossess dfferent altes and whether they reeve rvate sgnals. Therefore, or reslts do not rely on the exstene of asymmetr nformaton and/or dfferental altes, t deend only the onvexty of omensaton n relatve erformane and the exstene of sgnfant areer onerns. Graham 999 onsders a model atterned after Sharfsten and Sten 990 to nvestgate the herdng ehavor of nvestment newsletters. Treman 994 also rooses a model that demonstrates herdng y analysts.

7 Usng data from the IBES hstory taes on ndvdal analysts foreasts etween 988 and 000, we test the redtons of or model. Sefally, eah qarter we rank analysts nto deles ased on ast foreastng erformane allated over varyng tme horzons that range from year to 3 years. For eah analyst, we allate oldness as the normalzed devaton from the onsenss as n Hong, Kk, and Solomon 000. We then omte medan ftre oldness for eah dele of ast relatve erformane. Consstent wth or redtons, we fnd evdene of a U-shaed relaton etween ftre oldness and ast erformane. It s ossle that ths U-shaed relaton s a reslt of analysts ersstng n hoosng old or onservatve strateges and not de to dynam alteraton of strateges y analysts. In other words, an analyst who s always old s more lkely to take on extreme ostons n relatve erformane. We ontrol for ths osslty n or emral tests, and fnd that the U-shaed attern stll holds. As an alternatve methodology, we test for the U-shaed relaton sng the Fama-MaBeth 973 regresson methodology. These tests lead to qaltatvely smlar onlsons. Or emral reslts ontrte to the exstng lteratre n that we nvestgate the foreastng ehavor of all analysts n resonse to ther ror erformane and doment, for the frst tme, a non-monoton U-shaed relatonsh etween the oldness of an analyst's foreast and her ror relatve erformane. Hlary and Menzly 00 also emrally nvestgate the relatonsh etween oldness and ast erformane. Consstent wth or fndngs, they reort that analysts who have erformed well n the ast tend to sse older foreasts. However, they do not doment that analysts who have erformed oorly n the ast also sse older foreasts. They roose a ehavoral model ased on overonfdene to exlan ther fndngs. Or model offers a ratonal exlanaton for old foreastng ehavor y ot-erformers and nder-erformers. Hong et al 00 emrally examne the effet of areer onerns on oldness and fnd that more

8 exerened analysts are more lkely to devate from the onsenss. From an emral standont, we omlement ther fndngs y examnng and domentng the effet of ror erformane on analysts nentves to devate from the onsenss. 3 The remander of the aer roeeds as follow. In Seton we resent the model. In Seton 3, we state or man reslts regardng the otmal foreastng ehavor of analysts. In Seton 4, we resent the reslts of emral tests of or redtons. Seton 5 onldes the aer. All detaled roofs are relegated to the Aendx.. The Model The rmary fos of ths aer s the dynam foreastng ehavor of an analyst n resonse to hs ror erformane relatve to hs eers. We onsder an nfnte horzon, mlt-erod framework. The set of foreastng dates s haraterzed y the set Γ {..., 3,,,0,,,3,... }. The set of foreastng dates s a doly nfnte set to emhasze the ont that there s no ntal date n or framework. The fndamental eonom varale we model s the mlatve foreastng erformane, that s, the sm of the foreastng errors, for an analyst over a tme horzon T. T s the exogenosly sefed horzon over whh the analyst s erformane s evalated. The atal vale of T does not lay an mortant role n or analyss and t may also dffer aross analysts. 4 If e t denotes the mlatve foreastng error of an analyst at date t Γ, and e m t denotes the medan mlatve foreastng error for all analysts over the same tme horzon t, then q t e t e t m 3 H et al. 003 fnd a smlar U-shaed relaton etween relatve rsk hoes of mtal fnd managers n resonse to ther ror relatve erformane. 4 Emrally, we allow T vary etween one and three years. Ths does not qaltatvely alter the reslts.

9 denotes the relatve foreastng erformane of the analyst at date t. q. s a stohast roess wth sort n,. The medan foreastng error s sed as a enhmark rely for onreteness. Or theoretal reslts do not deend on the sef enhmark sed. At any date t, we assme that an analyst has the hoe etween adotng a old strategy or a onservatve strategy. If he adots the old onservatve strategy, then the hange n hs relatve foreastng erformane over the next erod s a normally dstrted random varale wth mean and standard devaton wth >. Therefore, q t q t N q t q t N nder the old and onservatve strateges and resetvely. In the aove, N s a standard normal random varale. 5 It s mortant to emhasze here that the old and onservatve strateges may reresent ether rvate or lly oservale sgnals. In other words, or theoretal framework does not rely on any sef assmtons regardng the oservalty of the analysts sgnals on the ass of whh they make ther foreasts. Therefore, asymmetr nformaton s not a ral ngredent of or model althogh or model s ertanly onsstent wth t. Frther, we do not make any sef assmtons regardng the altes of analysts, that s, they may all ossess the same alty or have dfferent altes. The fat that nether asymmetr nformaton nor dfferental analyst altes lays a ral role n or theoretal analyss makes t sgnfantly dfferent from earler theoretal frameworks that have een aled to examne analyst ehavor. 6 In all these frameworks, asymmetr nformaton and/or dfferental analyst altes lay mortant roles. In smmary, the natre of the nformaton that analysts ossess and the analysts foreastng altes do not affet 5 Sne the analyst an only hoose ether the old or the onservatve strategy at any date, we se the same notaton for the standard normal random varale wthot loss of generalty. 6 See, for examle, Sharfsten and Sten 99, Zweel 995, Prendergast and Stole 996, and Graham 999

10 or model as well as or theoretal reslts. Ors s a smle ratonal model of foreastng strategy hoes y analysts. There s an exogenosly sefed level of relatve erformane q sh that f q. exeeds q, the analyst s fred wth some roalty α 0,. 7 We also assme that the analyst ears sgnfant ersonal osts from eng fred. These roortonal osts are desred y the arameter δ [0,], that s, the analyst loses a roorton δ of her exeted ftre omensaton s he s fred. The analyst s assmed to e rsk-netral and hs omensaton s assmed to e onvex n hs relatve erformane. The rsk-netralty of the analyst s assmed rely for smlty of exoston and does not affet or reslts qaltatvely. For analytal tratalty, we assme that the omensaton of the analyst at date t s gven y 3 C t [ g ex q t] where g > 0, > 0. For sseqent notatonal smlty, we normalze g to. From 3, we note that as the analyst s erformane vares from eng very good to very ad, q t vares from eng sgnfantly negatve to eomng sgnfantly ostve. The goal of the rsk-netral analyst s to hoose hs foreastng oly ξ reresentng hs hoe etween a old and onservatve strategy at every foreastng date, n order to maxmze hs dsonted exeted omensaton. At any date t, hs otmal ftre foreastng oly shold therefore solve 4 v q t s s ξ ξ E[ δ v q τ E[ δ v q τ τ s t e s τ s t e s ex q C s] ξ ξ s] 7 The assmton that the analyst s not fred wth ertanty s onsstent wth the emral fndngs of Hong, Kk, and Solomon 000. They fnd that the roalty of an analyst gettng fred nreases wth ndererformane, t t s sgnfantly dfferent from one.

11 In the aove, τ denotes the random date at whh the analyst s fred. Note that the analyst s ayoff when he s fred s a roortonal of hs otmal vale fnton v q τ that s hs maxmm exeted ftre omensaton. We have nororated the fat that the analyst s otmal oles are learly statonary, that s, hs deson at any date deends only on hs relatve erformane at that date. The ssrts on q, C denote the deendene of the analyst s relatve ξ ξ erformane and omensaton on the foreastng oly he hooses. s the analyst s onstant dsont rate for ftre ash flows. We an se standard dynam rogrammng tehnqes to otan the followng Hamlton-Jao-Bellman eqaton for the analyst s otmal vale fnton: q > q v q s, e E[ v q ex q fred δv q ] In the aove, q reresents the analyst s relatve erformane at the end of the next erod f he follows strategy and hs relatve erformane at the egnnng of the erod s q. The frst term n the exetaton s the otmal vale fnton at the end of the erod, the seond term s the analyst s omensaton for the erod, and the thrd term reresents the roortonal ersonal osts he ays f he s fred de to hs relatve erformane eng aove the threshold level q. The ssrt fred reresents the event that the analyst s fred. Inororatng the exogenos roalty α that the analyst s fred when hs relatve erformane s aove the threshold, we otan v q s, e E[ v q ex q αδ v q q > q ]

12 The Contnos Tme Framework For analytal tratalty, we now make the standard ontnos tme aroxmaton that s vald when the tme erod etween sessve dates s small relatve to the tme erod over whh the analyst s foreastng ehavor s eng nvestgated. In ths ase, eqaton for the evolton of the analyst s relatve erformane s relaed y 5 dq t dt db t nder strategy {,} The analyst s ojetve n 4 s relaed y where B. s a Brownan moton. 6 v q t s s ξ ξ E[ δ v q τ E[ δ v q τ τ t dse s τ t dse s ex q C s] ξ ξ s] For sseqent analytal and notatonal onvenene, we re-defne the analyst s otmzaton rolem n terms of the roess t ex q t. Usng Ito s lemma, the evolton of the roess. nder strategy {,} s gven y 7 d t t[ dt db t] and the analyst s ojetve s to hoose hs foreastng oly to maxmze 8 t s E[ δ τ dse s ] ξ τ t s ξ where. s the analyst s otmal vale fnton as a fnton of q e. Note now that low hgh vales of. reresent good ad relatve erformanes y the analyst. In order to ensre that the analyst s vale fnton s defned, we assme that >,,. If. s the otmal vale fnton of the dynam otmzaton rolem 8, then, as mentoned earler, the analyst s otmal oly s statonary, that s, hs hoe of strategy at any

13 date s not exltly deendent on tme and deends only on the rrent measre of hs relatve erformane. We denote the analyst s otmal vale fnton y. We may se tradtonal dynam rogrammng argments analogos to those sed n dervng the dsrete-tme Hamlton- Jao-Bellman eqaton to wrte down the followng formal Hamlton-Jao-Bellman eqaton for : ] [ s 0, αδ > where. In the dynam rogrammng framework, the varale aove reresents the vale of the state varale. P so that the term s the nstantaneos rate of omensaton of the analyst. Hene, n regons where strategy s otmal, the vale fnton mst satsfy the system of ordnary dfferental eqatons: > 0; 0; αδ It an e shown that the general solton to the ordnary dfferental eqatons has the form: 9 D C B A > ; ; αδ ρ ρ where, and ρ ρ, are the ostve and negatve roots resetvely of the qadrat eqatons:

14 0 x x x 0 x αδ 0 In order to ensre that the analyst s vale fnton exsts, we assme that her dsont rate > and s hgh enogh so that ρ,. The followng lemma ollets roertes of the roots,, ρ, ρ that wll e sed freqently. emma a, ρ ρ, ρ, ρ Proof. In the Aendx. We now state wthot roof the followng well-known verfaton theorem for the analyst s otmal vale fnton. Prooston : Sose s a fnton that s dfferentale on 0, and twe dfferentale on { } 0, \ satsfyng the HJB eqaton 0 s,[ > αδ ] and lm / no les ondton. Then s the analyst s otmal vale fnton. Proof. See Karatzas and Shreve [998]. Ths omletes the formlaton of the model and the mathematal relmnares. 3. The Analyst s Otmal Foreastng Poly In ths seton, we exltly derve the otmal foreastng oles for the analyst for all ossle ars of old and onservatve strateges and haraterzed y the volatlty arameters

15 , wth > and hs dsont rate s hgh enogh to ensre that ondton s satsfed. We show that there exst two levels l, h wth l h of ror relatve erformane as measred y the roess. sh that t s otmal for the analyst to hoose the onservatve strategy when hs ror erformane les etween the thresholds and swth to the old strategy aove the threshold h and elow the threshold l. Therefore, the analyst hooses the old strategy when he s ether a sgnfant ot erformer or nder erformer and the onservatve strategy when he s an ntermedate erformer. We may have n whh ase the analyst l h always hooses the old strategy and the swthng of strateges s s-otmal. We rovde a neessary and sffent ondton for the swthng of strateges to e otmal for the analyst. The ntton nderlyng these reslts s the followng. If the analyst s a sgnfant ot erformer, the rsk of hs gettng fred s very low. Therefore, he hooses the old strategy sne t nreases hs exeted omensaton. On the other hand, f the analyst sgnfantly nder erforms the threshold where he may e fred, he faes sgnfant rsk of eng fred. He therefore, hooses the old strategy to maxmze the roalty that hs erformane may nrease aove the level elow. At ntermedate levels of erformane, the analyst trades off the hgher exeted omensaton from hoosng the old strategy for the lower emloyment rsk from hoosng the onservatve strategy. In general, there exsts an ntermedate regon of relatve erformane where the analyst refers the onservatve strategy. We now roeed to formalze ths ntton. Consder the lass of oles defned y the trgger r wth r where the analyst always hooses strategy for, r and strategy for r. It an e shown that the vale fnton r of sh a oly has the followng fntonal form:

16 ; ; ; r D r C B A r r r r r αδ ρ where the oeffents are determned y ontnty and dfferentalty ondtons at the onts r, and ther deendene on the trgger r s exltly ndated. Defne the dfferental oerators, ;, ' as follows: ' ; αδ The followng roostons omletely haraterze the otmal oles for the analyst. Prooston : Sose 3 0 > There exst a threshold level of relatve erformane > sh that f a,, t s otmal for the analyst to hoose the old strategy and f, t s otmal for the analyst to hoose the onservatve strategy. Proof. In the Aendx. Condton 3 of the rooston s therefore a sffent ondton for the analyst to hoose the onservatve strategy n some regon of ror relatve erformane. Inttvely, the ondton exresses the fat that the emloyment rsk of the analyst and the dfferene etween the rsks of the old and onservatve strateges are hgh enogh to ensre that t s otmal for the analyst to hoose

17 the onservatve strategy when hs ror erformane s lose to the threshold. The followng rooston shows that ondton 3 s also neessary for the otmalty of swthng to the onservatve strategy n some regon of ror erformane. Prooston 3. Sose 4 0 Then the otmal oly for the analyst s to always hoose the old strategy. Proof. In the Aendx. If ondton 4 s satsfed, the emloyment rsk of the analyst and/or the dfferene n the rsks of the two strateges s low enogh that t s s-otmal for the analyst to devate from the old strategy. The reslts of the aove roostons mly that t s ether always otmal for the analyst to hoose strategy or there exsts a non-emty ntermedate regon [, ] of ror erformane where the analyst otmally hooses strategy. If the analyst hooses strategy, he devates less from the medan analyst than f he hooses strategy. Sose now that strategy reresents a rvate nosy sgnal for the analyst and strategy reresents the herdng strategy. In ths settng, or reslts mly that when the analyst s a sgnfant ot erformer or nder erformer, he trsts hs rvate sgnal whereas f he s an ntermedate erformer, he hooses to herd. We wold lke to emhasze agan that or model does not make any sef assmtons aot whether the analyst reeves rvate sgnals. Hene, or reslt that sgnfant ot erformers and nder erformers devate more sgnfantly n ther foreasts from the medan than ntermedate erformers does not deend on whether there s asymmetr nformaton and/or the analysts ossess dfferent foreastng altes. Or model and reslts deend on two man assmtons: an analyst s omensaton s onvex n her relatve erformane and faes sgnfant areer onerns.

18 4. Emral Reslts Data Desrton: In order to test the aove roostons on ast foreast aray and oldness, we ollet foreasts from the IBES Detaled Hstory dataase over the erod 988 to 000. The Detaled Hstory dataase traks the dentty of the analyst ssng the foreast, her emloyer, the date of the foreast, and the atal vale of her foreast. Ths dataset also allows s to dentfy and trak eah analyst aross tme, even f they swth nvestment. In order to onstrt or measres of foreast aray and foreast oldness, we se the rankng methodology ntroded n Hong, Kk, and Solomon 000. The roedre s as follows. We se the I/B/E/S data to onstrt a qarterly erformane measre ased on an analyst s foreast aray. We defne F,j,t as the most reent earnng-er-share foreast of qarterly earnngs ssed y analyst on stok j n qarter t. Or measre of analyst s aray for frm j n year t s the asolte dfferene etween her foreast and the realzed earnngs-er-share of the frm, A j,t : foreast error F A,, j, t j t We then sort the analysts who over a frm n a qarter ased on ther foreast errors gven aove. We then assgn a rankng ased on ths sortng: the est analyst reeves a rank of one, the seond est analyst reeves a rank of two, and so on. In the ase of tes, we assgn eah analyst the mdont vale of the ranks that they take. Sne the maxmm rank an analyst an reeve for a frm deends on the nmer of analysts who over the frm, we sale an analyst s rank y the nmer of analysts who over the frm. The formla for ths sore measre s: 00 rank, j, t aray sore, j, t 00, nmer analysts j, t

19 where nmer of analysts j,t s the nmer of analysts who over the frm n a gven qarter. 8 We then allate the average sore for eah analyst over the revos for, eght, and qarters. Hgher overall sores orresond to etter analyst erformane. We se a smlar roedre to onstrt a measre of an analyst s foreast oldness. et n F, j, t m F m, j, t, where s the set of all analysts other than analyst who rode an earnngs estmate for stok j n year t, and n s the nmer of analysts n. Hene, F, j, t s a measre of the onsenss foreast made y all other analysts exet analyst followng stok j n qarter t. oldness F F,, j, t, j, t, j t We then relate the revos rankng methodology for onstrtng the analyst aray sore as n the revos sseton. Emral Fndngs Tale resents smmary statsts for or samle of analysts foreasts. Or samle ontans a large nmer of analysts from a nmer of dfferent nvestment frms. The average nmer of nqe nvestment frms eah qarter s , whle the average nmer of nqe analysts ssng foreasts eah qarter s, The average analyst n or samle sses qarterly earnngs-er-share estmates for 7.3 stoks. 9 The average stok n a samle has 5.56 dfferent analysts rovdng overage. Note that we reqre at least two analysts to e overng the stok n order to allate or oldness and erformane rankngs. The average analyst n or samle has 8 For examle, the lowest rated analyst for eah frm wold reeve a sore of zero, whle the hghest rated analyst wold reeve a sore of The maxmm nmer of stoks overed y an analyst s 94. Ths old e attrted to team of analysts rather than an ndvdal.

20 4.46 years of exerene. Fnally, average analyst oldness and average analyst aray average and By onstrton, the medan vales of oth of these varales s Tale examnes whether analysts atvely move etween onservatve and old foreastng strateges. The tale shows a transton matrx relatng average ast oldness to ftre oldness. The tale s onstrted as follows. For eah qarter n or samle, we allate the average oldness sore for eah analyst over the revos for qarters and then dvde analysts nto deles ased on ths sore. We smlarly rank analyst nto oldness deles ased on ther rrent oldness sore. The reslts sggest that analysts atvely move etween old and onservatve strateges. For examle, of the analysts ranked nto the lowest ast oldness dele, only 5.93% rse the least old strategy n the sseqent erod. Smlarly, of the analyst ranked nto the hghest oldness dele ased on ast erformane, only 8.0% fall nto the same dele n the sseqent erod. It s worth notng that the -vale from a χ test ndates that we an rejet the nll hyothess of eqal roortons wthn eah ast oldness dele. Tale 3 resents or reslts on the relaton etween ast erformane and ftre oldness. Eah qarter, we rank analysts nto deles ased on ther average ast erformane. The average ast erformane of eah analyst s allated sng aray sores over the revos for, eght, and twelve qarters. We then omte mean ftre oldness for eah of these deles. In order to test for a U-shaed relaton, we omter the average dfferene n oldness etween ast erformane deles 5 and 6 and then test to see whether mean oldness n eah dele s dfferent from ths vale. Panel A resents or fndngs for the ase where ast erformane s allated over the revos for qarters. The reslts onfrm the exstene of a U-shaed relaton. The worst ast erformane dele has ftre oldness of 5.06, whh s statstally sgnfant. Smlarly the est ast erformane dele has ftre oldness of 50.6, whh s statstally

21 sgnfant. The reslts n Panel B and Panel C show that the reslts are not affeted y the hoe of the tme horzon over whh ast foreast aray s allate. Smlar reslts otan f we allate average ast erformane sng the revos eght qarters or the revos twelve qarters. In Tale 4, we ontrol for the ast oldness of the analyst. It s ossle that the U-shaed relaton domented n Tale 3 s a reslt of analysts ersstng n hoosng old or onservatve strateges and not de to dynam alteraton of strateges y analysts. In other words, an analyst who s always old s more lkely to take on extreme ostons n relatve erformane. The reslts resented n Tale 4 anel are stll onsstent wth a U-shaed relaton etween ast erformane and ftre oldness even after ontrollng for ast oldness. In ontrast to the reslts resented n Tale 3, we fnd that the relaton tends to e drven y the est ast erformane dele and the worst ast erformane dele. Panels B and C show that the hoe of tme horzon over whh ast foreast aray s allated rodes smlar reslts. Fama-MaBeth Regressons: Or fndngs aove are onsstent wth a U-shaed relaton etween oldness and ast erformane. In ths seton, we se Fama-MaBeth 973 regressons to examne the rostness of or reslts. For every qarter n or samle erod, we estmate a ross-setonal regresson relatng ftre oldness to ast aray. Gven the reslts n Tale 4, we nlde n the regresson model ndator varales f the analyst s ast erformane s ether n the to erformane dele or the ottom erformane dele. Based on the exstng lteratre, we also ontrol for an analyst s exerene and the nmer of frms overed y the analyst. Hgh exerene s an ndator varale takng the vale of one f the analyst has more than for years of ror exerene and zero

22 otherwse. Nmer of frms overed s the nmer of frms the analyst overs n a gven qarter. We also ontrol for the average oldness of the analyst over the revos for qarters. In Tale 5, we reort the average oeffents from these regressons along wth the assoated -vale from a smle t-test for the statstal sgnfane of the estmates. In sefaton, we fnd that analysts n the worst ast erformane dele and analysts n the est erformane dele have sgnfantly hgher oldness than other analysts. Sefaton shows that ths reslt holds even after ontrollng for exerene, ast oldness, and the nmer of frms overed y the analyst. Interestngly, the oeffent on nmer of frms overed s negatve and sgnfant, ndatng that analysts overng more stoks tend to sse more onservatve foreasts. The fnal sefaton examnes the nteraton etween ast erformane and exerene and oldness. The reslts ndate that exerened analysts are more lkely to devate from the onsenss followng oor erformane. However, exerened analysts wth good ast erformane are not more lkely to sser older foreasts. 5. Conlsons We examne the dynam foreastng ehavor of nvestment analysts n resonse to ther ror erformane relatve to ther eers wthn a ontnos tme/mlt-erod framework. The model dffers sgnfantly from exstng theoretal models n ts dynam natre and the fat that t does not rely on any sef assmtons regardng the exstene of asymmetr nformaton and/or dfferental analyst altes. The entral redton of the model s that there s a U-shaed relatonsh etween the oldness of an analyst's foreast, that s, the devaton of her foreast from the onsenss and her ror relatve erformane. In other words, analysts who sgnfantly ot erform or nder erform ther eers sse older foreasts than ntermedate erformers.

23 We then test the redtons of or model on oserved analyst foreast data. Consstent wth or theoretal redtons, we doment an aroxmately U-shaed relatonsh etween the devaton of analysts foreasts from the onsenss and ther ror relatve erformane. Ths reslt s rost to dfferent emral methodologes. Consstent wth ror emral evdene, or reslts mly that analysts who fae greater emloyment rsk have greater nentves to herd. Or theoretal and emral analyses therefore hghlght the mortane of oth exlt nentves n the form of omensaton strtres and mlt nentves n the form of areer onerns, on the dynam foreastng ehavor of analysts. As noted y Welh 000 one drawak of many exstng models of herdng s that they are stat and desgned only to exlan a steady state n whh all analysts herd erfetly. These models are nale to exlan why analysts may devate from the onsenss some tmes and herd at other tmes. We address ths sse y theoretally and emrally examnng the dynam foreastng ehavor of nvestment analysts n resonse to ther ror erformane relatve to ther eers.

24 Referenes Fama, E. and J. MaBeth, 973, Rsk, retrn, and eqlrm: Emral tests, Jornal of Poltal Eonomy 8: Graham, John, 999, Herdng among nvestment newsletters: Theory and evdene, Jornal of Fnane 54: Hlary, G. and. Menzly, 00, Does ast sess lead analysts to eome overonfdent? Workng aer: Unversty of Chago. Hong, H., J. Kk, and A. Solomon, 000, Serty analysts areer onerns and herdng of earnngs foreasts, Rand Jornal of Eonoms 3: -44. H, P., J. Kale, and A. Sramanan, 003, Relatve rsk hoe y mtal fnd managers, Workng aer: Georga Teh. Jao, J., T. ys, and M. Neale, 999, Exertse n foreastng erformane of serty analysts, Jornal of Aontng and Eonoms 8: 5-8. Karatzas, I. And S. Shreve, 998, Methods of Mathematal Fnane, Srnger-Verlag: New York. eone, Andrew and Joanna W, 00, What does t take to eome a serstar? Evdene from nstttonal nvestor rankngs of fnanal analysts, workng aer, Unversty of Rohester. Mkhal, M., B. Walther, and R. Wlls, 999, Does foreast aray matter to serty analysts? The Aontng Revew 74: Morrs, S, 997, An nstrmental theory of oltal orretness, Workng aer: Unversty of Pennsylvana. Prendergast, C. and. Stole, 996, Imetos yongsters and jaded oldtmers: Aqrng a retaton for learnng, Jornal of Poltal Eonomy 996: Rosen, S., 98, The eonoms of serstars, The Ameran Eonom Revew 7: Sharfsten, D. and J. Sten, 990, Herd ehavor and nvestment, The Ameran Eonom Revew 80: Welh, I, 000, Herdng among serty analysts, Jornal of Fnanal Eonoms 58: Treman, B., 994, Analyst foreasts and herdng ehavor, Revew of Fnanal Stdes 7: 97-4

25 Zweel, J., 995, Cororate onservatsm and relatve omensaton, Jornal of Poltal Eonomy 03, -5:

26 Aendx Proof of emma We frst note that 0 sne y assmton. Therefore, the aove mles that sne, are the roots of the qadrat eqaton 0 x x. We have 0 > sne,, > > >. Therefore, 0 > It follows that mst e greater than,,.e.. We may smlarly show that ρ ρ. We now note that 0 αδ αδ sne the exresson n the rakets on the rght hand sde aove s zero y the defnton of the root and 0 > αδ. From the aove, t easly follows that we mst have ρ ρ. Proof of Prooston : The roof roeeds y exltly onstrtng a dfferentale fnton that satsfes the hyotheses of Prooston, that s, A 0 ] [ s ', > and / lm

27 We frst show that there exsts wth sh that the fnton s twe dfferentale everywhere exet ossly at and that A 0;, r We egn y notng that the fnton r s a ontnos fnton of r. Condton 3 of the rooston therefore mles that there exsts > ' sh that A3 0 ' ' > By the defnton of the fnton A4 0 and ' ' A5 0 ' ' ' settng Stratng A4 from A3, we see that sne ' d d [ ' ' ' ' ] > 0 d d ' r n, we see that ' s dfferentale everywhere y onstrton. Ths mles that ' d d A6 [ ' ' ' ' ] > 0 d d A5 and A6 learly mly that A7 0 ' ' We now show the exstene of sh that A8 0

28 It s not dfflt to show that ths mles that s twe dfferentale at and that A9 0 We rove ths y frst showng that A0 r r r lm As r, the vale fnton r learly aroahes the vale fnton of the oly of hoosng strategy for >. It s easy to see that the fntonal form of for > s A C We now note that A ] ] [ [ lm lm C Sne 0, the lmt of the frst term aove s zero. Sne 0 as, > and 0 as, the seond term s ostve and goes to nfnty as. Ths mles that A0 holds 0. It now easly follows y ontnty that there exsts > sh that A8 holds and therefore A9 holds. We now show that s the reqred otmal swthng ont where s defned y settng r n. By the reslt of Prooston 3, we need to show that 0 Strtly ths needs to e shown rgorosly, t the argments are qte straghtforward and are avalale on reqest.

29 A3 ' for 0; for 0; for 0; For >, A4 D ] [ Sne y hyothess, 0. Sne y defnton, 0 > as >. Therefore, the seond term on the rght hand sde aove s always negatve. Therefore, A9 an hold only f the frst term on the rght hand sde of A4 s ostve and the seond term s negatve. Sne 0, t follows that the exresson D ] [ s a dereasng fnton of. Therefore, A9 mles that A5 for 0 > Usng the fat that s twe dfferentale at, we an show after some tedos algera that we omt for the sake of revty that A5 mles that the oeffents, C B n the defnton of satsfy A6 0 0, > C B We now note that

30 A7 C B ] [ Sne 0, t follows from A6 that the frst and seond terms on the rght hand sde aove are oth nreasng fntons of. Sne 0, the thrd term aove s ostve and therefore also an nreasng fnton of. Therefore, s an nreasng fnton of for. A8 and A9 now learly mly that A8 for 0 It only remans to show that A9 0 ' for 0 A8 mles that the vale fnton of the analyst mst e strtly greater than the vale fnton of hoosng strategy for all vales of. The vale fnton of hoosng strategy always mst e at least as great as αδ that s the vale the analyst otans from hoosng strategy when he faes emloyment rsk for all vales of, that s, there s no fnte threshold eyond whh the analyst does not fae emloyment rsk. From the defnton of, ths easly mles that we mst have A0 0 > A We now note that

31 A ] [ ' αδ αδ ρ ρ ρ A for Sne,, ρ ρ ρ >, A0 mles that the frst term on the rght hand sde aove s negatve and the seond term s also negatve. Ths learly mles A9. Therefore, we have shown that the vale fnton satsfes the hyotheses of Prooston 3 and s therefore the otmal vale fnton of the analyst. Hene, the oly of swthng oles at s otmal. Ths omletes the roof. Proof of Prooston 3 Settng r n, the vale fnton has the followng fntonal form: A ; ; D A > αδ ρ Sne mst e at least as great as the vale fnton of hoosng strategy when the analyst faes emloyment rsk for all vales of, we mst have A3 0 > A However, the vale fnton s strtly less than the vale fnton of hoosng strategy when the analyst faes no emloyment rsk. Therefore, we mst have

32 A4 0 D In order to estalsh the otmalty of hoosng strategy for all vales of, we need to show that A5 > 0, 0, ' We now note that A6 ] [ ' αδ αδ ρ ρ ρ A for Sne,, ρ ρ ρ >, A3 mles that the frst term on the rght hand sde aove s negatve and the seond term s also negatve. Therefore,, 0 '. For >, we have A7 D ] [ If, then A4 mles that the frst term on the rght hand sde aove s negatve. Sne, the seond term s also negatve. On the other hand, f >, then the frst term on the rght hand sde of A7 s ostve and the seond term s negatve. Ths mles that s a dereasng fnton of for >. Hyothess 4 of the rooston now mles that >, 0. Therefore, we have estalshed A5. Hene, the fnton satsfes

33 the hyotheses of Prooston 3. Hene, the oly of always hoosng strategy s otmal. Ths omletes the roof.

34 Tale. Smmary Statsts Ths tale shows varos smmary statsts for or samle of analysts foreasts. The data onssts of all qarterly foreasts of earnngs er share etween 988 and 000 ontaned n the IBES Detal Hstory Fle. The nmer of nvestment frms eah qarter and nmer of analysts ssng foreasts eah qarter are the average nmer of nvestment frms smttng foreasts to IBES eah qarter and the nmer of nqe analysts smttng foreasts resetvely. Exerene s allated for eah analyst as the dfferene etween the year of the foreast and the analyst s frst year smttng foreasts to the IBES dataase. Analyst oldness and analyst aray are allated sng the rankng roedre of Hong, Kk, and Solomon 000. Mean Medan Std. Devaton Mn Max Nmer of nvestment frms eah qarter Nmer of analysts ssng foreasts eah qarter,763.67, ,86.00 Nmer of stoks overed er qarter Nmer of analysts followng a stok Exerene Average analyst oldness Average analyst aray

35 Tale. Transton Matrx Ths tale shows a transton matrx relatng ast average oldness to ftre oldness. The tale s onstrted as follows. For eah qarter n or samle erod, we rank analysts nto deles ased on ther average oldness sore over the revos for qarters. We erform a smlar rankng of the oldness sores for the rrent sore. The -vale from a χ test for eqal roortons s reorted for eah ast oldness dele. Average oldness over revos 4 qarters Boldness east Bold Most Bold -vale from χ test for eqal roortons east Bold 5.93%.09% 8.43% 7.6% 7.44% 0.46% 6.94% 7.58% 9.50% 5.0% %.90%.90% 0.80% 0.07% 9.87% 9.49% 8.85% 8.34% 8.00% %.53%.7%.59%.05% 9.74% 0.% 9.45% 8.47% 6.97% % 0.79%.5%.8% 0.98% 0.%.05% 0.06% 8.6% 6.68% % 9.96%.04%.76%.56% 9.77%.49% 0.9% 9.4% 6.5% % 9.45% 9.78% 0.3% 0.45%.46% 0.6% 0.9% 9.3% 8.89% % 9.48% 9.59% 0.9% 0.99% 9.97%.05%.76% 0.86% 8.0% % 8.80% 9.6% 0.7% 9.48% 0.7% 0.96%.6%.07% 9.5% % 8.8% 7.85% 8.9% 8.96% 9.76% 0.90%.9%.7%.37% 0.00 Most Bold 4.97% 8.6% 7.07% 6.55% 7.0% 0.54% 7.39% 8.67%.33% 8.0% 0.00

36 Tale 3. Past erformane and ftre oldness Ths tale examnes the relaton etween ast erformane and ftre oldness. Analysts are ranked eah qarter on the ass of ther average aray over the revos for, eght, and twelve qarters. The analysts are then sorted nto deles, wth Dele ontanng the worst erformng analysts and Dele 0 ontanng the est erformng analyst. We then omte mean ftre oldness for eah dele. The -vale reorted for eah dele tests whether the mean oldness s statstally dfferent from the average oldness of qartles fve and sx. Panel A: Performane rankngs ased on one year of ast data Past Performane qartle Past Performane Boldness # of oservatons P-vale Worst Best Panel B: Performane rankngs ased on two years of ast data Past Performane qartle Past Performane Boldness # of oservatons P-vale Worst Best Panel C: Performane rankngs ased on three years of ast data Past Performane qartle Past Performane Boldness # of oservatons P-vale Worst Best

37 Tale 4. Past Performane and ftre oldness ontrollng for ast oldness Ths tale examnes the relaton etween ast erformane and ftre oldness, ontrollng for ast oldness. Past oldness s allated as the average oldness sore for eah analyst over the revos for qarters. The -vale reorted for eah dele tests whether the mean oldness s statstally dfferent from the average oldness of qartles fve and sx. Panel A: Performane rankngs ased on one year of ast data. Past Performane qartle Boldness # of oservatons P-vale Worst Best Panel B: Performane rankngs ased on two years of ast data Past Performane qartle Boldness # of oservatons P-vale Worst Best Panel C: Performane rankngs ased on three years of ast data Past Performane qartle Boldness # of oservatons P-vale Worst Best

38 Tale 5. Fama-MaBeth Regressons Ths tale reorts the reslts of Fama-MaBeth regressons. The deendent varale n eah regresson s the oldness sore. Worst ast erformane dele s an ndator varale that takes the vale of one f the analyst s average foreastng erformane over the revos for qarters was n the ottom 0% and zero otherwse. Best ast erformane dele s an ndator varale that takes the vale of one f the analyst s average foreastng erformane over the revos for qarters was n the to 0% and zero otherwse. Hgh exerene s an ndator varale that takes the vale of one f the analysts has for or more years of exerene and zero otherwse. Nmer of frms overed s the nmer of frms overed y the analyst n the qarter. Past oldness s the average oldness sore of the analyst over the revos for qarters. We nlde ontrols for ndstry effets. We se the IBES SIG ode to defne the ndstres. The average R from the 5 qarterly regressons are reorted. P-vales are reorted n arentheses. Boldness 3 Interet Worst ast erformane dele Best ast erformane dele hgh exerene ast oldness nmer of frms overed Worst ast erformane delehgh ex Best ast erformane delehgh ex Indstry Effets YES YES YES N Average R