Trading on Shor-Term Informaion Preliminary version. Commens welcome Alexander Gümbel * Deparmen of Economics European Universiy Insiue Badia Fiesolana 5006 San Domenico di Fiesole (FI) Ialy e-mail: guembel@daacomm.iue.i 3 May, 999 * I wish o hank James Dow, my hesis supervisor, for very helpful advice. I have benefied from discussions wih Paolo Baigalli, Caherine Casamaa, Rainer Kiefer, Thomas Marioi, Jean-Charles Roche, Rober Waldmann, Lucy Whie and Wilfried Zanmann. I wish o hank he Universiy of Toulouse, where par of his research was carried ou, for heir hospialiy. Financial Suppor from he European Invesmen Bank is graefully acknowledged. All remaining errors are mine.
Absrac In his paper we address he quesion as o why fund managers may rade on shor-erm informaion in a financial marke ha offers more profiable rading on long-erm informaion. We consider a seing in which a fund manager s abiliy is unknown and an invesor uses performance observaions o learn abou his abiliy. We show ha an invesor learns less efficienly abou he abiliy of a fund manager when he rades on longerm informaion compared o rading on shor-erm informaion. This is he case, because he informaion on which a manager bases his rades is less precise he longer he informaion horizon, and hus performance observaions conain more noise. Moreover, under rading on long-erm informaion, performance observaions become available afer a shor period only if he manager unwinds his posiion early. Such performance observaions, however, are generally conaminaed wih addiional noise, because unwinding prices only reveal underlying asse value imperfecly. When he informaional efficiency of shor-erm prices increases, his effec becomes less pronounced, because a long-erm rader who unwinds his posiion afer a shor ime can convey an increasing amoun of informaion concerning his abiliy o he invesor. A he same ime, rading on shor-erm informaion becomes less profiable, and herefore he invesor s incenive o induce shor-erm rading weakened. Neverheless, we show ha shor-erm rading may be induced even when prices fully reveal shor-erm informaion. Journal of Economic Lieraure Classificaion Numbers: D8, D83, G4, G3. Keywords: Managerial abiliy, learning, delegaed porfolio managemen, shor-ermism, price efficiency.
. Inroducion There has been considerable debae among economiss and praciioners alike, concerning shor-ermism in financial markes. In his debae, shor-ermism by fund managers is frequenly held responsible for he mispricing of long-erm asses and he resuling underinvesmen by firms in such asses. This, in urn, is alleged o resul in low growh raes of shor-ermis economies. Shor-ermism refers o a siuaion in which facors concerning he near fuure carry an excessive weigh in decision making compared o facors regarding he longer erm. Excessive is here defined relaive o a firs-bes benchmark prevalen in a fricionless economy. In he conex of financial markes, shor-ermism is ypically undersood o mean ha invesors or raders pu oo much emphasis on shor-erm informaion, such as shor-erm profis and cash-flows, when valuing an asse. In his paper we argue ha incomplee informaion concerning he fund manager s inheren abiliy, may lead invesors o prefer a deviaion from he firs-bes informaion horizon, resuling in rade on shor-erm informaion as a second bes oucome. In conras o much of he exising lieraure on shor-ermism, we explore he role of he rading horizon in allowing an invesor o learn abou he unknown abiliy of a fund manager. One of he reasons for shor-ermism frequenly pu forward is ha fund managers ac under inense shor-erm pressures, leading hem o neglec longer-erm objecives (see Marsh, 990, Froo, Scharfsein, and Sein, 99, Dow and Goron, 994). A number of heoreical conribuions have shown ha due o agency problems, firm or fund managers may indeed ake a decision ha exhibis a shor-erm bias, alhough his is undesirable from an invesor s poin of view. While agency problems associaed wih fund and firm managemen may share a number of commonalies, one has o disinguish clearly beween hese wo seings when aemping o explain shor-ermism. Firm managers have he choice of invesmen projecs which may pay off in he more or less disan fuure. Inefficien invesmen in shor-erm projecs may occur, because a manager canno convey his abiliy quickly o an invesor by choosing he long-erm projec (v.thadden, 995). A major difference beween a firm s invesmen decision and a fund s porfolio choice is ha here is ypically no (or only a very illiquid) marke for invesmen projecs. Therefore, no marke price for invesmen projecs exiss, which makes i hard for an ousider o assess he value of such projecs See Marsh (990) for a comprehensive appraisal of his debae.
before heir payou dae. As a resul, a firm manager canno signal superior invesmen skill in he shor-run by selling a long-erm invesmen projec. Insead, he migh choose an invesmen projec ha pays off in he shor run. In conras o his, he marke value of a porfolio of liquid securiies changes over ime as addiional informaion ges incorporaed ino prices. A porfolio manager who rades on long-erm informaion can hus signal superior abiliy early on, by unwinding such a long-erm posiion afer a shor period, when prices reflec more of he informaion on which he originally based his rade. Therefore, i is no obvious ha he ofen cied shor-erm pressures under which fund managers ac, are a saisfacory explanaion for shor invesmen horizons. As Demirag (995) wries: I is... reasonable o argue ha pressures o maximise shor-run reurns... are in principle compaible wih a willingness o ignore shor-erm cash flows, profis, and dividends in favour of long-erm prospecs. A fund manager who consisenly recognised such prospecs and invesed accordingly, shorly before ohers did, would perform exremely well in he shor-erm. In his paper we consider a seing in which he price of an asse becomes informaionally more efficien as is liquidaion dae is approached and a fund manager can eiher rade in an asse ha is liquidaed in he near fuure (rade on shor-erm informaion) or he more disan fuure (rade on long-erm informaion). If he rades on long-erm informaion he can unwind his posiion afer a shor period. The invesor who is unaware of he manager s inheren abiliy, uses he informaion conained in a manager s pas performance, in order o learn abou his abiliy and possibly o swich funds o a differen manager if his performance is bad. 3 In his conex we show ha rade on shorerm informaion may be preferred by he invesor, because i allows her o learn more efficienly abou he abiliy of he manager. This is he case, alhough rade on long-erm informaion would be chosen in he firs-bes benchmark case. Trading on long-erm informaion leads o less efficien learning abou he manager s abiliy for wo reasons. Firsly, he qualiy of informaion on which he manager rades, worsens as he ime For shor-ermism by firm managers see for example Narayanan (985), Sein (989), Shleifer and Vishny (990) and von Thadden (995). For shor-erm biases by fund managers due o agency problems see Shleifer and Vishny (997). These papers are reviewed in more deail in Secion 6F. 3 This may happen in he form of individual invesors swiching ou of badly performing and ino well performing funds or by funds firing heir manager afer bad performance. Empirical evidence suggess ha boh of his is happening. For evidence on fund swiching see Chevalier and Ellison (997) and for firing of fund managers, see Chevalier and Ellison (998) and Khorana (996).
horizon of informaion increases. This reflecs he idea ha i is easier for a fund manager o predic an even in he near fuure han in he disan fuure. Trading on less precise informaion, however, implies ha performance observaions conain less informaion abou he abiliy of he manager, because bad performance is more likely o be aribuable o bad luck, raher han low abiliy. Therefore, rading on long-erm informaion allows less efficien learning abou a manager s abiliy compared o rading on shor-erm informaion. Secondly, when a manager rades on long-erm informaion and unwinds his posiion early, he price achieved hrough unwinding is iself a garbled signal of he underlying liquidaion value of he asse. This adds a furher layer of noise o he performance observaions available when rade occurs on long-erm informaion. In our seing early unwinding of long-erm posiions is never inferior o a buy-and hold sraegy. Expeced rading profis are he same under eiher sraegy (buyand-hold or early unwinding), bu under a buy-and-hold sraegy, performance observaions become available laer, which is undesirable for he invesor. The efficiency of learning ha resuls from observing a long-erm rader s shor-run performance clearly depends on he shor-erm informaional efficiency of prices. When prices become perfecly informaive a he ime of unwinding (shorly before he asse is liquidaed), observing he manager s shor run performance is as informaive as observing he acual asse value. Since he manager s rading decision is based on his assessmen of underlying asse value, he invesor can bes judge he performance of he manager, when prices reveal mos informaion a he ime of unwinding, i.e. when prices are mos informaionally efficien a an inerim dae before liquidaion. The less informaionally efficien hese shor-erm prices are, he harder i becomes for he invesor o learn from performance observaions. The shor-erm informaional efficiency of prices affecs he principal s choice of rading horizon for anoher reason. The profiabiliy of rading on shor-erm informaion depends on he degree o which informaion is incorporaed ino he price upon submission of he manager s order. When more informaion is incorporaed ino his price, rading becomes less profiable. Therefore, an increase in shor-erm informaional efficiency reduces he principal s payoff from inducing rade on shor-erm informaion. 4 We hus esablish a link beween an invesor s incenive o induce rading on long-erm informaion and he shor-erm informaional efficiency of prices. 4 This decline in he economic value of informaion corresponds o findings of Grinold (997) who demonsraes ha he profiabiliy of rading on a paricular piece of informaion decreases as he dae of public revelaion of he informaion draws nearer. I also corresponds o he reamen of Dow and Goron (994) and Vives (995), where raders 3
We show ha even in he exreme case when prices fully reveal shor-erm informaion, an invesor may wish o induce shor-erm rading in a manager of unknown abiliy as his allows more efficien screening. The remainder of he paper is organised as follows. Secion lays ou he basic model. In Secion 3 he expeced profis are derived for long-erm and shor-erm rading, and he firs-bes oucome is calculaed. In Secion 4 i is shown ha he payoff o he invesor is increasing wih he degree of shor-erm informaional efficiency of prices when rade occurs on long-erm informaion. Secion 5 gives he main resul concerning he desirabiliy of rading on shor-erm versus long-erm informaion. Secion 6 is a discussion of he resuls and Secion 7 concludes. The Appendix conains he proofs.. The model We consider a seing in which here is one invesor ha hires a fund manager ha can eiher acquire long-erm or shor-erm informaion. Depending on his ype, he qualiy of his informaion is high or low. Afer acquiring informaion he manager can rade in risky securiies, where rades are execued by a marke maker. As in Kyle (985) he marke maker is in Berrand compeiion wih oher marke makers and herefore ses a price ha is equal o he expeced discouned value of he securiy, given oal order flow. Toal order flow consiss of he order submied by one informed rader (he fund manager) and an order submied by noise raders. Trades are (opimally) unwound afer one period and he resuling prices and profis are observable by he invesor who uses his informaion o updae her belief concerning he manager s abiliy. Subsequenly, he invesor decides wheher o reain he manager for anoher rading period, or o fire he manager and hire a new manager from a pool of indisinguishable ypes. We model a financial marke in discree ime wih infiniely many daes T = {0,,,...}. A each dae here is a riskless securiy wih rae of reurn r and wo risky securiies k {A, B }. The risky securiy k pays an uncerain dividend d k, {0,} only once, a dae, and no dividend a any oher dae. Eiher realisaion of he dividend paymen is equally probable and independen of he oher securiies dividend paymens. 5 subsequenly rade on informaion concerning a paricular poin in ime, making prices more efficien as he even dae is approached. 5 Insead of dividend paymens one could also hink of he uncerain payoff as he liquidaion value of an asse. 4
There are wo ypes of agens in he economy. A risk neural invesor who delegaes porfolio managemen o a risk neural manager wih limied liabiliy. The manager can acquire informaion abou he uncerain dividend paymen of securiies of he same vinage, i.e. informaion ha concerns he dividend paymen of wo risky securiies a he same dae. In paricular, a dae, a manager can acquire a noisy signal for dividend paymens one period from now (d A,+, d B,+ ) or wo periods from now (d A,+, d B,+ ). This choice is denoed by a {a s,a l }. For a = a l, he manager receives a long-erm signal l {DD, DU, UD, UU} a dae for d A,+, d B,+. In his signal, D sands for down (low dividend realisaion) and U for up (high dividend realisaion).the firs leer in he signal indicaes he dividend for asse A +, while he second indicaes ha for asse B +. For a = a s he manager receives a shor-erm signal s {DD, DU, UD, UU } for d A,+, d B,+. A any dae he manager can acquire one of he wo signals a zero cos, while i is prohibiively cosly o acquire boh signals a he same ime. 6 There are wo ypes of fund manager m {L, H} and neiher he principal nor managers know he ype. Depending on his ype, he informaion acquired by a manager is of differen qualiy. In paricular, if a manager is a high ype, he signal l (s ) is correc for boh asses of ha vinage wih probabiliy µ (ν), is correc for one asse bu incorrec for he oher wih probabiliy µ ( ν ), and is never incorrec for boh asses. For a low ype manager on he oher hand, a signal is always correc for one asse and incorrec for he oher, and i is unknown for which asse i is correc. Moreover, i is assumed ha ν>µ, i.e. i is harder o predic dividends wo periods ino he fuure han one period ino he fuure. Tables and below show he probabiliies of a paricular realisaion of dividends (in he columns) condiional on a paricular long-erm signal received (in he rows) for a high and a low ype manager. The analogous disribuion applies o shor-erm signals, where µ is replaced by ν everywhere and (d A,+, d B,+ ) is replaced by (d A,+, d B,+ ). One way o hink abou he link beween rue asse value and signal received is he following. Fund managers ofen acquire informaion concerning boh specific asses and general economic condiions ha affec he value of asses. In our seing one could undersand he manager as acquiring informaion concerning a paricular ime horizon, such as he ineres rae se by he cenral bank in say six monhs ime. He hen ries o undersand how a paricular value of ha 6 Since we are concerned wih he problem of a choice of ime horizon here, we allow managers o choose he ime horizon of heir informaion, while no addressing he issue of a choice of a paricular asse (A or B). 5
l DD µ (-µ)/ (-µ)/ 0 l DD 0 / / 0 ineres rae will affec a large number of asses in he economy, where differen asses are affeced differenly. One could hen hink of he low ype manager as being unable o inerpre his informaion correcly in a consisen manner. He herefore rades some of he asses in he correc and some ohers in he wrong direcion. Wih a large number of asses, his amouns o pracically always levelling ou he number of wrong and he number of correc invesmens, which is exacly wha happens in his simplified wo asse economy. (d A,+, d B,+ ) (0,0) (0,) (,0) (,) DU (-µ)/ µ 0 (-µ)/ UD (-µ)/ 0 µ (-µ)/ UU 0 (-µ)/ (-µ)/ µ Table : The high ype s probabiliy of receiving a paricular signal is given depending on he underlying sae of naure. (d A,+, d B,+ ) (0,0) (0,) (,0) (,) DU / 0 0 / UD / 0 0 / UU 0 / / 0 Table : The low ype s probabiliy of receiving a paricular signal is given depending on he underlying sae of naure. The high ype manager on he oher hand, is someimes able o rade more han half he asses in he correc direcion (when he is lucky), while someimes he performs badly and rades some correcly and some no. His probabiliy of rading more han half he asses in he correc direcion depends on he informaion horizon, reflecing he idea ha i is harder o predic far away evens correcly han evens in he nearer fuure. Noe ha alhough he model exhibis more han one asse ha may pay a dividend a any given dae, his paper is no concerned wih issues such as diversificaion across asses. Inroducing more han one asse allows us o model he evoluion of a manager s repuaion over ime in a 6
paricularly simple manner. This allows us o obain analyical soluions o a problem ha is only racable numerically in a more general seing. There is an infiniely large pool of managers and all agens have he correc prior γha a randomly seleced manager is a high ype (m=h). 7 The principal can decide a any dae wheher or no o reain he presen manager. Denoe his choice by e {0,}, where e = means ha a dae he manager is reained from he previous period. If a manager is fired (e =0) he principal picks a new manager a random from he pool wihou incurring any coss. Denoe by m he ype of manager ha is employed a dae afer he employmen decision e has been aken. Moreover, denoe by u he probabiliy ha a manager is a high ype jus before he employmen decision e is aken and q he probabiliy jus afer. This probabiliy is also referred o as he repuaion of a manager and depends on he managers employmen and performance record. The fund manager receives a privae benefi b in every period he is employed. Doing so is a shor-hand way of saying ha he receives a consan wage paymen every period, so ha he prefers being employed by a fund over no being employed. Such a consan wage paymen corresponds o mos conracs found in real world arrangemens, where he manager is ypically rewarded on he basis of ne asse value under managemen. This ype of wage conrac yields incenives mainly implicily, as invesors may wihdraw funds from he manager when performance is bad. On he oher hand, a manager whose performance is good, will ypically be able o arac more funds and hus increase wage paymens. This corresponds o he firing/reainmen decision of he invesor in our model. Moreover, he srucure of he model is sufficienly simple o ensure ha a manager will ake he bes rading decision for he invesor, merely because he wishes o remain employed. Therefore, we do no need o consider he provision of incenives hrough complicaed wage conracs. 8 Following Kyle (985), we model he financial marke as being informaionally semi-srong efficien. Therefore, we assume ha beside he informed raders here are also noise raders who have an exogenous demand for he securiy (e.g. an unmodelled hedging need). A dae hey submi 7 One could hink of he fund managers in he pool as agens wihou work experience. I hen seems plausible o assume ha hey do no ye know how ap hey are for he job of a fund manager. 8 Papers exploring he provision of incenives hrough opimal wage conracs for delegaed porfolio managers include Bhaacharya and Pfleiderer (985), Soughon (993) and Heinkel and Soughon (994). 7
a random and serially uncorrelaed order n { n n} ~, k, τ, for asse k τ, where τ {+, +}. 9 Eiher realisaion of n k, τ, is equally probable and independen of d k,τ. We assume ha noise raders hold heir posiions unil he dae of he uncerain dividend paymen. 0 A dae he marke maker receives a oal order for asse k τ, denoed by Q = n + θ k, τ, k, τ, k, τ, where θ k, τ, denoes he (marke) order submied by he informed rader. Marke makers are in Berrand compeiion, and herefore make zero profis in expecaion. The price p k, τ, for asse k τ a dae is herefore se so as o equal he asse s expeced presen value given he marke maker's informaion se I. Hence, p k,, τ = /(+r) τ- E[ d k,τ I ]. Since we are ineresed in exploring a financial marke ha exhibis more profiable longerm han shor-erm rading opporuniies, while preserving he naural propery ha long-erm informaion is no beer han shor-erm informaion, we require ha prices become informaionally more efficien as he even dae draws nearer. This is achieved by assuming ha for each securiy k + he marke maker receives a noisy shor-erm signal w k, {0,} a dae abou d k,+. The informaion conen of he signal is defined as ω prob(d k,+ = w k, = ) = prob(d k,+ = 0 w k, = 0) /. Anoher way of achieving increased informaional efficiency of shor-erm prices would be o inroduce a second informed rader who exogenously rades on shor-erm informaion. This would leave he main insighs of he paper unchanged, while complicaing he analysis considerably, which is why his approach was no aken here. All he signals l, s +, and w + are assumed o exhibi minimal correlaion, so ha prob(w A, =x, w B, =y, s =X d A,+,d B,+ ) = prob(w A, =x d A,+ ) prob(w B, =y d B,+ )prob(s =X d A,+,d B,+ ), 9 Noe ha he acual order size n is irrelevan here as i depends enirely on he scale used for measuring order size. Wihou loss of generaliy we can se n=. We will do his laer when calculaing rading profis, bu for he ime being we reain he noaion in order o avoid confusion wih oher variables. 0 For a discussion concerning he behaviour of noise raders and heir role in our model, see Secion 6.C and 6.D. In order o explain why a long-erm arbirage opporuniy may remain unexploied, we need o consider a siuaion in which a long-erm arbirage opporuniy should be exploied in a firs-bes seing (oherwise shor-ermism would no be an issue). The assumpion is essenially made, so ha a siuaion can arise in which i is no firs-bes o rade on shor-erm informaion. 8
where x,y {0,} and X {DD, DU, UD, UU}. Similarly for he long-erm signal: prob(w A,+ =x, w B,+ =y, l =X d A,+,d B,+ ) = prob(w A,+ =x d A,+ ) prob(w B,+ =y d B,+ )prob(l =X d A,+,d B,+ ), Moreover we assume ha shor-erm and long-erm signals display minimal correlaion: prob(s + =Y, l =X d A,+,d B,+ ) = prob(s + =Y d A,+, d B,+ ) prob(l =X d A,+,d B,+ ). I is assumed ha he marke maker knows he manager s repuaion, denoed by q. The marke maker s informaion se a a given dae hus consiss of he observed oal order flow in all asses, his privae signal w k, and he choice of a rading horizon a, which can be inferred from observed order flows. Hence, he informaion se is I = { Q, τ,, Q, τ,, a, q, w A,, w B, }. A B Since noise raders submi buy or sell orders of size n, he informed rader has o submi orders of he same size (θ τ, {-n, 0, n}), as any oher order size would cerainly reveal he manager s order o he marke maker. Since he marke maker knows ha he manager only submis a buy (sell) order afer receiving a signal ha indicaes ha he dividend paymen for ha asse will be high (low), he price would be se so as o reflec his informaion and rading could no be profiable. We moreover assume ha he manager can unwind a long-erm posiion before a new round of rade begins. This yields an unwinding price a + for an asse k +, denoed by P k,+ which is used by he principal o updae her belief abou he manager s ype. For simpliciy i is assumed ha he invesor can conrac on he ype of he signal ha he manager acquires. Given ha a any dae she can coslessly fire he curren fund manager and pick a new manager from he pool, he agency problem reduces o a conflic of ineres concerning he decision of he invesor o reain he presen manager, or o fire him and hire a new manager. 3 Since he manager receives a privae benefi of being employed and has limied liabiliy, he always wishes o be employed. Therefore, we can simply se he wage paymen o he manager equal o zero in I is sraighforward o specify he marke maker s ou of equilibrium beliefs for oal order sizes oher han -n, 0, n, such ha i is no profiable for he manager o deviae from equilibrium. For a reamen of his issue see Dow and Goron (994), Secion VI. 9
every period. The invesor only wishes o employ he manager if she is a leas as well off wih him as wih a randomly seleced manager from he pool. Summary able of variables k {A, B } d k, {0,} a {a s, a l } l s Securiy ha pays off uncerain dividend a dae Dividend paymen of securiy k Horizon of informaion acquired a dae Signal when long-erm informaion is acquired Signal when shor-erm informaion is acquired n w k, Marke maker s signal for d k,+ µ Probabiliy ha long-erm signal is correc for boh asses ν Probabiliy ha shor-erm signal is correc for boh asses ω Probabiliy ha w k, is correc m {L, H} Type of manager rading a dae e {0,} Invesor s employmen decision (for e =0 he manager is fired) q u { n n} k, τ,, θ k τ Repuaion of manager afer employmen decision Repuaion of manager before employmen decision Order for asse k τ submied by noise rader a dae,, Order for asse k τ submied by informed rader a dae Q k, τ, p k, τ, P k, Toal order for asse k τ submied a dae Price for asse k τ a dae Unwinding price for asse k (a dae -) Table 3: Summarises he variables of he model. 3 Some models of career concerns and learning abou an agen s ype (e.g. Holmsrom, 98) assume ha an agen s wage flucuaes wih his repuaion, rendering he principal indifferen beween reaining and firing he agen. This approach assumes ha agens have all he bargaining power, i.e. ha he labour marke is no compeiive. Our approach is compaible wih a compeiive labour marke and finds empirical suppor e.g. in Chevalier and Ellison 0
The iming of evens. The holders of securiy k receive a dividend d k,.. The marke maker observes w k, abou d k,+. 3. If he informed rader held a long-erm posiion in asse k + (θ k,+,- 0), he can unwind he posiion. 4. The principal updaes her belief abou he ype of manager employed from observing d k, and/or he profi due o unwinding. 5. The principal akes a firing/ reainmen decision e. 6. The principal chooses a new rading horizon a. 7. The manager observes a signal s if a =a s (l if a =a l ). 8. The manager submis an order θ k,+, (θ k,+, ) and noise raders submi n k,+, (n k,+, ). 9. The marke maker observes Q k,+, (Q k,+, ), ses prices p k,+, (p k,+, ) and rades are execued. 0. Resar a. 4 Define π + (a, q ) W + (a, q ) - W (+r) as he rading profi ha accrues when a manager of repuaion q rades on informaion a raher han invesing all wealh in he riskless securiy. The principal maximises he expeced presen value of fuure wealh subjec o he sochasic ransiion of he repuaion of he manager employed. The evoluion of repuaion depends on he principal s acions and is capured by consrains (), (3) and (4) of he opimisaion problem below. A dae : [ ( ( ) + ( + ( ) + )] V ( η u r E a q u e V a a e u, ) = max π,, η,,, ~ a, e + () u if s.. q (u,e ) = γ if ( ) e e = () = 0 u~ + ~ h η, a, q (3) η + = if a = a = al, e = 0 oherwise (4) (998) and Khorana (996). Boh papers find ha a fund manager s probabiliy of being fired is negaively correlaed wih pas performance.
If he manager ges fired (e =0), he repuaion of he following manager will jus be γ, as managers from he pool are picked a random. If he manager is reained, his nex period repuaion is a random variable described by he disribuion funcion h(*). The sochasic properies of nex period s repuaion depend on he previous repuaion q, he rading horizon a induced and he realisaion of he indicaor funcion η {0,}. The indicaor funcion is se o if he manager employed a dae - was also employed a - (e - =) and raded on long-erm informaion (a - =a - =a l ). The variable η - is imporan, because i capures he asymmery beween learning abou a manager who rades on long-erm informaion for he firs ime or repeaedly. Is significance will be discussed in deail in Secion 4 and Appendix B. 3. Asse prices and rading profis In his Secion resuls concerning he rading profis accruing from rade on shor-or longerm informaion are presened. The purpose of he secion is o illusrae how rading can be profiable in his seing and how he marke maker s privae informaion affecs rading profis. Moreover, Proposiion gives a necessary and sufficien condiion for rading on long-erm informaion o be firs-bes. 3. Asse prices and rading profis under shor-erm rading Firs, consider prices and rading profis under shor-erm rading (a =a s ). Since he manager submis orders of size n, oal order flow in each asse A + and B + can ake he values Q k,+, {- n, 0, n}. The manager can eiher receive a signal ha indicaes ha he dividend paymen for, say, asse A + will be high (i.e. s = UU or s = UD), in which case he submis a buy order (θ A,+, = n), or he receives a bad signal for asse A + (e.g. s = DU or s = DD) and sells he asse shor (θ A,+, = - n). 5 Orders for asse B + are deermined similarly. This leads o he possible oal order flows 4 Noe ha in his saemen of iming, a =0 he poins and 3-5 do no apply. 5 We assume ha γν /4, i.e. he manager rading on shor-erm informaion is mos likely o rade in he correc direcion when following his signal, raher han guessing or doing he opposie of wha he signal suggess. If he principal believes ha he manager does rade on his informaion in he way described above, he manager has no incenive o deviae from doing so, as his would only reduce his probabiliy of being reained. We do no consider possible equilibria in which he manager rades in he wrong direcion and he principal believes ha his is wha he is doing. For he generaliy of he argumen in he Theorem of Secion 5, we canno resric /4. However, all he argumens go hrough under eiher assumpion: ha he long-erm rader does or does no follow his signal when </4.
Q k,+, = n: Boh, he manager and he noise rader submi an order. Q k,+, = 0 : The manager submis a buy order and he noise rader a sell order, or vice versa. Q k,+, = -n: Boh manager and noise rader submi a sell order. Apar from he order flow, he marke maker also receives a direc signal w k, {0,} for he nex dividend paymen d k,+. Correspondingly, he price p k,+, depends on he oal order flow and he marke maker s privae signal. Prices can be calculaed by Bayesian updaing from p k,+, (I ) = /(+r) E[d k,+ I ] = /(+r)prob(d k,+ = I ) (5) Since he acual price of an asse depends on he realisaions of Q A,+,, Q B,+,, w A,, w B,, we ge 3 3 =36 possible prices for each asse. For he compuaion of rading profis, however, no all of hese prices are relevan, because he manager can only expec o make a profi if i so happens ha oal order flow does no reveal his order (i.e. when Q k,+, =0). These relevan prices are given in he proof of he following lemma, conained in Appendix A. Lemma : The expeced rading profis when a manager of repuaion q rades on shor-erm informaion a dae are given by: Eπ + (a =a s, q ) = q ν ω(-ω) (6) Proof see Appendix A. Noe ha expeced rading profis are decreasing in ω, for ω>/. This is he case, because an increase in he qualiy of he marke maker s privae informaion ω, resuls in informaionally more efficien prices. Hence, he manager s informaional rens from rading decrease. In he limiing case when he marke maker has perfec informaion abou nex period s dividend paymens (ω=), prices fully reflec his informaion and he manager makes zero profis. 3. Asse prices and rading profis under long-erm rading Nex, consider he price seing behaviour of he marke maker, when he manager rades on long-erm informaion a dae, i.e. a =a l. In ha case he submis an order θ k,+, for asse k +. Again, oal order flow in each asse can ake he values -n, 0, or n. However, when rading on long-erm 3
informaion, he manager submis an order for asses for which he marke maker has no ye received privae informaion. Hence, he price only depends on he realisaions of Q k,+,, which implies nine possible differen prices for each asse. Again, he price of asse k + a dae (denoed by p k,+, ) can be calculaed by Bayesian updaing. Since an asse pays ou a mos one dividend, we can wrie p k,+, (I ) = /(+r) E[d k,+ I ] = /(+r) prob(d k,+ = I ) (7) Afer one period he manager can unwind his posiion wih he marke maker, afer he laer received his privae signal w k,+, bu before he nex round of rading begins. As in he case of shorerm rading his implies 36 differen possible unwinding prices. The proof of Lemma in Appendix A conains he deails of how prices are formed. We can now sae he following resul concerning he rading profis under long-erm informaion acquisiion. Lemma : Suppose a manager of repuaion q rades on long-erm informaion a dae and unwinds he posiion a +. Expeced rading profis are hen given by: Eπ + (a =a l, q ) = q + µ ( r). (8) Moreover, expeced discouned rading profis from following a buy and hold sraegy are he same as under he unwinding sraegy. Proof see Appendix A. I is imporan o noice ha by observing he prices a which he manager unwinds his posiions, he principal learns he realisaion of he marke maker s privae informaion. This in urn is a noisy signal for he rue value of he fuure dividend paymens d k,+, which is used by he principal o assess wheher or no he manager raded in he correc direcion a dae. This is imporan for he principal s decision concerning he choice of an informaion horizon for her fund manager. 6 6 Noe ha he qualiy ω of he marke maker s signal w does no affec he expeced profiabiliy of rading under longerm informaion, alhough he individual prices do depend on ω. This is he case because an increase in ω leads o an increase in expeced profis, when rades are unwound a a favourable price. This increase in expeced profis is exacly mached ex ane by a decrease in expeced profis, when rades are unwound a an unfavourable price. As a resul expeced profis are independen of ω. 4
3..3 The firs-bes benchmark Consider as a benchmark he case where he invesor is able o disinguish high and low ype managers and hus employs a high ype. Proposiion : For µ > µ * 4νω( ω )( ) + r (9) a high ype manager rades more profiably in expecaion when acquiring he long-erm signal a l han when acquiring he shor-erm signal a s. Moreover, if and only if /4 > ω(-ω)(+r), (0) is i possible o find parameer values ν, µ, r, such ha ν > µ > µ *, i.e. rading on long-erm informaion is more profiable even hough he long-erm signal is less informaive han he shorerm signal. Proof see Appendix A. In he remainder of he paper we are mainly ineresed in he case where ν > µ > µ *. Noe ha when ω = /, condiion (0) can never be saisfied. This is obvious, since for ν>µ, shor-erm informaion is beer han long-erm informaion, and a ω = /, shor-erm prices are inrinsically no more informaionally efficien han long-erm prices. As a resul rading on shor-erm informaion is always firs-bes. For all values of ω > /, here exiss an r small enough such ha (0) is saisfied. In he exreme case, where ω =, he condiion is saisfied for all values of r <. 4. Informaional efficiency of shor-erm prices and rading on long-erm informaion In he previous secion i was shown ha he direc profi from rading on long-erm informaion is independen of he informaional efficiency of shor-erm prices, denoed by ω. Is i herefore he case ha he principal s payoff when he manager rades on long-erm informaion is independen of ω? Noe ha he principal s payoff consiss no only of he direc rading profi, bu also of he benefi from learning abou he manager s abiliy. When a manager rades on long-erm informaion a dae and unwinds his posiions a +, he manager learns he marke maker s privae informaion w A,+ and w B,+ by observing he prices a which rades are unwound. Since w A,+ and 5
w B,+ are indicaive of he rue value of he securiies A +, B +, he principal receives some informaion abou wheher or no he manager raded in he correc direcion, before he rue asse value is revealed. I is inuiively clear ha when ω increases, i.e. unwinding prices become a more reliable source of informaion for rue asse value, he principal is beer able o assess he manager s performance. Hence, one would expec he principal s payoff from inducing rade on long-erm informaion o be non-decreasing in ω. More formally, denoe by W l (η,u ) he principal s discouned expeced payoff from always inducing long-erm rade, when he currenly employed manager (before employmen decision e is aken) has repuaion u and an opimal employmen decision is aken a every dae from onwards. The sae variable η denoes wheher or no a manager who rades on long-erm informaion did so for he firs ime (η =0) or no (η =). To see why his is imporan consider he following. e + = η + = R=(w +,l +,d + ) η + =0 R=(w,l ) R=relevan informaion e + =0 η + =0 R=(w +,l + ) + + Figure shows he evoluion of he sae variable η depending on he employmen decision e under long-erm informaion acquisiion (i.e. a =a l and a + =a l ) and he associaed relevan informaion o carry ou he belief updae abou a manager s ype. If a manager is fired a + (e + =0) he dividend paymen d + is uninformaive abou he new manager s ype (R=(w +, l + )). When a manager is reained a dae + and raded on long-erm informaion a dae (η + =), d + is informaive abou his ype (R=(w +, l +, d + )). A dae + he principal receives he following informaion: l (by observing he posiions θ k,+, ha were aken), w k,+ (by observing a which prices P k,+,+ posiions are unwound) and acual dividend paymens d k,+. Signal l and w k,+ are direcly informaive abou he abiliy of he manager since he marke maker s signal w k,+ is informaive abou nex period s dividend paymens d k,+ and hus abou wheher or no he manager raded asses A + and B + in he correc direcion a dae. On he oher hand, d k,+ reveals wheher or no rades wo periods ago (rade θ k,+,- a dae - 6
) were correc. This, however, is only of ineres o he principal if he manager who raded in - is sill employed. This allows us o wrie he expeced discouned payoff under he opimal employmen decision as W ( η u ) l (, ) ( + r) q u e, = max e s.. consrains (), (3), and (4) µ [ ( η ( ), ~ ( )] + + r l + + E W e u e () Depending on he value of ω, he opimal employmen policy may differ and hence he maximised expeced discouned payoff. Denoe by W l * (ω, η, u) he maximised payoff for a given value of ω, η and u. Then, we can sae he following resul: Proposiion : The principal s expeced discouned payoff W l * (ω, 0, γ) when always inducing longerm rading and choosing an opimal employmen policy is non-decreasing in ω, and for γ< sricly increasing in ω for some value ω * [/,]. Proof see Appendix B. To see why his is rue, consider he exreme case where ω = /, i.e. he marke maker receives no privae informaion. In ha case he unwinding prices P k,+ do no reveal any informaion abou he fuure dividend ha was no already conained in he previous price. Therefore, he principal does no learn anyhing abou wheher or no he manager received a correc long-erm signal from observing he manager s rading profis and insead has o wai unil he dividend paymen acually occurs. For a newly employed manager his means ha he firs informaion abou his abiliy is observed wo periods afer he is firs employed. The principal hus has o reain a poenially bad manager for a leas one more period han under perfecly efficien shor-erm prices (ω = ). I is essenially his delay in learning ha causes long-erm rading o become less aracive as he informaional efficiency of shor-erm prices decreases. 7
5. Shor-erm versus long-erm rading In his Secion he main resul is presened, followed by a discussion of is driving forces. Some implicaions of he resul are explored. The proof is conained in Appendix B. Our main ineres in his paper is o find ou wheher or no shor-erm rading may be induced wih a manager picked randomly from he pool. We are ineresed in he se of parameer values r, µ, ν, γ, for which his may be he case, in paricular when rading on long-erm informaion is firs-bes. Denoe he se on which he basic parameers are defined by B={(r, µ, ν, γ) (r, µ, ν, γ) {R + [/,] [0,]}, γν /4}. Denoe by A(ω) B he se for which shor-erm rading is induced when a manager picked from he pool is firs employed. Moreover, using Proposiion, we can denoe he se of parameers for which long-erm rading is firs-bes, while he shor-erm signal is more informaive, by F(ω)={(r, µ, ν, γ) (r, µ, ν, γ) B, ν>µ>µ * }. We would hen like o know wheher or no A(ω) is a non-empy se, how i depends on ω, and wheher we can have a siuaion where long-erm rading is firs-bes, while shor-erm rading is induced by he principal, i.e. F(ω) A(ω). All of his is saed in he following Theorem. Theorem: (i) For all values of ω (/,], here exiss a non-empy se A(ω) B of parameer values (r, µ, ν, γ), such ha he principal prefers o induce rading on shor-erm informaion wih a manager who is randomly picked from he pool. This is he case even when rading on long-erm informaion is firs-bes, i.e. A(ω) F(ω). Moreover, an approximaion for A(ω) F(ω) can be given by AF (ω) A(ω) F(ω) wih ( )( r) ( )( r) ν µ µ 4νω ω + AF (ω)={(r, µ, ν, γ) (r, µ, ν, γ) F(ω), > r + γν µ 4γνω ω + (ii) A(ω ) A(ω ) for ω >ω. }. () Proof see Appendix B. The Theorem above saes he following. () Even when long-erm rading is firs-bes, he principal may wan o induce a newly employed manager o rade on shor-erm informaion, for any degree ω>/ of he informaional efficiency of shor-erm prices. 8
() A sufficien condiion on he parameer values for shor-erm rading o be chosen by he invesor who employs a new manager, when long-erm rading is firs-bes, is given by (). (3) The higher he informaional efficiency ω of shor-erm prices, he lower he incenive for he principal o induce shor-erm rading. Hence, he se of parameers A(ω) for which shor-erm rading is induced by he principal becomes smaller as ω increases. In order o illusrae he mechanism a work in his model, i is mos convenien o consider he case where ω=, i.e. prices are fully revealing one period before he uncerainy concerning dividend paymens is resolved. In his case he evoluion of repuaion akes a simple form. Remember ha when a manager rades on long-erm informaion a dae (a =a l ), he unwinds his posiions a +, a prices P k,+,+. As menioned above, unless ω = /, P k,+,+ reveals he marke maker s privae signals w k,+. For ω =, he realisaion of w k,+ is perfecly informaive abou d k,+. By indirecly observing w k,+ a dae + he principal knows wheher or no he manager received a correc long-erm signal l in he previous period. As a resul, he repuaion updae u + can ake wo values: a high value if he manager received a correc signal and a low value if he received a wrong signal. From Bayesian updaing we ge he disribuion h(η, a =a l, q ) 7 as u ( ) q + = µ q µ wih probabiliy q µ wih probabiliy - q µ Under shor-erm rading, he principal observes direcly he relevan dividend paymens d k,+ a dae + and hus wheher or no he manager received a correc signal. Again, he repuaion updae for a reained manager can ake one of wo values: a high value if he signal was correc for boh asses and a low value if i was wrong for one asse and correc for he oher. We can hus characerise h(η, a =a s, q ) by (3) u ( ) q + = ν qν wih q ν wih - q ν (4) 7 For ω=, he variable η is irrelevan. 9
From (3) and (4) i is clear ha repuaion deerioraes afer bad performance. If a manager rades for he firs ime (and hence q = γ) and performs badly, his repuaion falls below he repuaion of managers in he pool. Since hiring and firing is cosless, he principal fires a manager whose repuaion is below ha of a manager picked from he pool. Therefore, a any poin in ime and under eiher rading sraegy, he principal employs a manager eiher of repuaion q= (if she was able o idenify a high ype manager) or of repuaion q=γ(if she has no ye been able o idenify a high ype manager). Under his opimal employmen policy, we can calculae he expeced repuaion E[q + q =γ,a ] for rading on shor- or long-erm informaion. I is easy o verify ha E[q + q =γ,a =a s ]=γ(+ν(-γ)) > γ(+µ(-γ))= E[q + q =γ,a =a l ] ν>µ and γ<. This is he case, because he screening value of a paricular rading horizon is direcly linked o he qualiy of informaion on which he manager rades. If a high abiliy manager rades on shorerm informaion he is more likely o receive a correc signal han when he rades on long-erm informaion. Since a manager can only be idenified as a high ype when he happens o receive a correc signal, he principal is more likely o become aware of a high ype manager s ideniy when she les him rade on shor-erm informaion. Thus, only when shor-erm informaion is of higher qualiy han long-erm informaion can he principal learn more efficienly from leing he manager rade on shor-erm informaion. An ineresing implicaion of our model is ha he sensiiviy of firing as a reacion o performance is sensiive o he manager s age. A young (newly employed) manager ges fired afer performing badly once. If he is reained he will never be fired alhough he migh perform badly in some periods. On an empirical level, his resul is suppored by Chevalier and Ellison (998). They find ha he probabiliy of a fund manager being fired afer bad performance decreases wih he manager s age. Anoher ineresing resul concerns he role of he cos of capial in deermining he opimaliy of shor-erm rading, which is saed in he following corollary. Corollary: When prices fully reveal shor-erm informaion, he principal wishes o induce rade on shor-erm informaion only if he opporuniy cos of capial r, is sufficienly low. 0
This resul conrass wih convenional wisdom (Marsh, 990) which associaes shor invesmen horizons wih srong discouning. Long-erm invesmen projecs pay off laer han shorerm projecs. When he discoun rae increases, a long-erm projec loses more in ne presen value han a shor-erm projec, leading o a shor-erm bias in he choice of invesmen horizon. The above corollary shows ha exacly he opposie holds in our seing for he special case ω=. From condiion () in he Theorem we can see ha for ω=, he riskless rae of reurn r mus be sufficienly small for shor-erm rading o be induced. The riskless rae r deermines he principal s (opporuniy) cos of capial and hus he rae a which fuure paymens are discouned. By inducing shor-erm raher han long-erm rading, he principal incurs an opporuniy cos of screening due o foregoing rading profis in he nex period. The gain from doing so only accrues in laer periods, as he principal learns more efficienly abou he abiliy of he manager employed. Therefore, only if ineres raes are sufficienly low, is a principal willing o induce shor-erm rading. More generally, he effec is no clear cu, as an increase in r does affec he expeced discouned rading profis under long-erm rading more srongly han hose under shor-erm rading (see equaions (6) and (8)). Therefore, in general, an increase in r has an ambiguous effec on he desirabiliy of shor-erm rading. The Theorem also shows ha rading on shor-erm informaion in a risky securiy may occur, even if i is equally profiable o inves in he riskless asse. This resul resembles Dow and Goron s (997) finding ha delegaed porfolio managers may churn, i.e. rade in a risky securiy, alhough his is no more profiable han rading in he riskless securiy. Our resul, however, is differen from oher churning resuls (e.g. Allen and Goron, 993) in ha our model exhibis rade in he risky asse ha is always based on he rader s informaion abou asse value. 6. Discussion of he resuls (A) Agens planning horizons In conras o mos of he lieraure on shor-ermism, we do no assume ha agens have exogenous limied horizons. We model all agens in he economy as having infiniely long horizons, which is imporan for wo reasons. Firsly, our specificaion is saionary, unlike oher models in he lieraure. E.g. v. Thadden (995) presens a model in which boh, principal and agen have a wo
period horizon. I is no obvious in such a model ha he agen s incenives and he principal s payoff (aking ino accoun he screening value of long-erm versus shor-erm projecs) remain unalered in an infinie horizon model. Secondly, our resuls are no driven by exogenous shor horizons as for example in Dow and Goron (994). In conras o heir resuls, we find ha shor-ermism may obain when all agens have infinie horizons. (B) Shor-ermism as a ransien feaure In our model shor-ermism occurs when a principal firs employs a manager, bu disappears once she has learned ha a manager is a high abiliy ype. The probabiliy of employing a manager under shor-erm rading decreases over ime and goes o zero as. One could herefore argue ha shor-erm rading occurs only in a very small number of periods compared o he ime in which long-erm rading occurs. I would be sraighforward o ge more shor-erm rading by assuming (realisically) ha managers had finie lives, or a consan probabiliy of separaion from he principal in each period. Then invesors would have o sar searching for a new manager in regular inervals and shor-erm rading would occur more ofen. Such a modelling approach, however, would have inroduced an elemen of limied horizons ha, as explained above, we wished o avoid. 8 (C) Exogenous liquidiy rade We presen a model in which he source of noise rade is exogenous. Some models using noise raders argue ha hey are irraional raders who paricipae in he sock marke despie making a loss consisenly due o mispercepions concerning asse value (see De Long e al., 990). Anoher way of modelling noise rade is o assume ha rade originaes from raional agens who face a wage shock ha is negaively correlaed wih asse value. For his reason hey submi orders for an asse, despie losing money on average. This approach is explored by Spiegel and Subrahmanyam (99) and used for example in Dow and Goron (994, 997). In principle, our model allows for he inroducion of raional agens ha ac as liquidiy raders because hey have a hedging need. This would, however, complicae he analysis somewha, because he orders submied by hese liquidiy raders will ypically depend on he repuaion of he informed rader, as he laer affecs he cos of insurance for he hedger. As a resul he model would 8 An alernaive modificaion migh be o le a manager s ype be non-consan over ime. This approach is followed by Benabou and Laroque (99) who find ha some paricipans in a financial marke do no find ou an informed rader s ype even asympoically, and herefore anomalies due o asymmeric informaion may persis infiniely.
become analyically less racable. Noneheless, we like o hink of our model as no essenially driven by he presence of irraional agens. (D) Early unwinding of long-erm posiions In order o model unwinding of long-erm posiions in a simple manner, we assume ha noise raders do no unwind heir posiions. Therefore, a marke maker knows ha only informed raders unwind heir posiions and hus does no need o infer from he previous price wheher a noise rader or an informed rader is unwinding a posiion. This simplifies he reamen considerably, because unwinding prices are hen uncondiional on he price in he previous period. I seems plausible ha noise raders hold heir posiions unil he dae of liquidaion, if heir demand for he asse originaes from a need o insure wage risk. If wage is correlaed wih he dividend paymen of an asse, i is clearly beer o wai unil he dividend paymen occurs han o unwind he posiion before his paymen occurs. We moreover assume ha unwinding of long-erm posiions is carried ou before new rades in he asse occur. In oher papers (e.g. Hirshleifer, Subrahmanyam, and Timan, 994, and Froo, Scharfsein, and Sein, 99) unwinding posiions occurs a he same ime as new orders for he asse are submied. This approach is no aken here, because i would imply ha a principal could only observe inerim performance of a long-erm rader, afer he nex rading round was compleed. Thus, firing a manager would impose a cos on he principal as she would have o wai for one period, before a new manager could paricipae in he sock marke. As a resul inducing long-erm rading for a manager of unknown abiliy would become even coslier, which reinforces our resul. Since we do no believe ha his effec is relevan in he real world, we preferred no o le i affec he resuls of he model. In our seing early unwinding of long-erm posiions is as profiable as keeping a posiion (long or shor) unil he liquidaion dae. Noneheless, he principal prefers early unwinding of he posiion, because his supplies her wih some informaion abou he manager s abiliy (for ω>/). Compared o shor-erm rading, he informaion conained in shor run performance is noneheless lower (for ω<). This addiional layer of noise under long-erm rading is due o he fac ha informaion ges refleced imperfecly in he unwinding price (depending on he parameer ω). In Hirshleifer, Subrahmanyam, and Timan (994) he unwinding decision by risk averse raders is also affeced by he informaion conained in unwinding prices. In heir model raders prefer o unwind a 3
posiion in an inerim period, because he inerim price does no ye reflec a public informaion shock. Hence, in conras o Hirshleifer e al., in our model early unwinding is preferred, because some of he public informaion is already conained in he unwinding price. (E) Incenive compaibiliy and he rading horizon We assume ha he rading horizon can be conraced upon. We do no need o assume his, as he prices would reveal o he principal which sraegy he manager followed, which makes he rading horizon enforceable (e.g. by hreaening o fire he manager if he should deviae from he agreed upon rading sraegy). As shown in v. Thadden (995) incenive compaibiliy problems arise when he principal wans o induce he long-erm sraegy. This is also he case in our seing. To see his, suppose ha he principal canno conrac on he rading horizon and is unable o infer from prices which rading horizon was chosen by he manager. If he principal wishes o induce rading on shor-erm informaion, she could simply offer a zero wage in every period. Since he manager receives a privae benefi from being employed he chooses he rading sraegy ha maximises he probabiliy of being reained. Consider he simple case in which shor-erm prices are perfecly efficien and he manager ges fired afer one bad performance. Then he probabiliy of being reained is under rading on long-erm informaion, and γν under rading on shor-erm informaion. By assumpion, ν>µ and hence he manager prefers o rade on shor-erm informaion. Under more complicaed employmen rules, essenially he same argumen applies, so ha generally rading on shor-erm informaion is incenive compaible, even if i is impossible o conrac on he rading horizon. A problem may arise if he principal wishes o induce long-erm rading wih an agen whose repuaion is lower han one (q<). Even under an appropriae incenive conrac i may hen no be opimal for he manager o rade on long-erm informaion. This, however, is exacly he case deal wih in v.thadden who shows ha shor-ermism may resul when i is no desired by he principal due o incenive compaibiliy problems. (F) Relaion o he lieraure Shor-ermism is ypically seen as undesirable, because of is adverse effec on he efficiency of sock markes and firms' invesmen decisions. Froo, Scharfsein, and Sein (99) for example show ha shor rading horizons may lead o serious mispricing of asses. Vives (995) finds ha 4
informaional efficiency of prices may increase or decrease under shor rading horizons, depending on wheher or no informaion arrival is concenraed or dispersed over ime. Dow and Goron (994) argue ha arbirageurs limied rading horizons may lead o a failure of long-erm arbirage and hus sock prices may no reflec he long run value of an asse. Shleifer and Vishny (997) poin o he failure of long-erm arbirage in he conex of delegaed porfolio managemen. 9 Regarding firms invesmen decisions, Shleifer and Vishny (990) argue ha if speculaors prefer shor-erm arbirage over long-erm arbirage, long-erm asses will be more srongly mispriced by he marke han shor-erm asses. This, in urn, may lead firm managers who are averse o mispricing, o underinves in long-erm asses. Similarly, v. Thadden (995), Sein (989) and Narayanan (985) argue ha firm managers may boos shor-erm earnings a he expense of long-erm earnings, if shor-erm performance affecs heir wage or repuaion. Shleifer and Vishny (997) focus on incenive problems for porfolio managers and consider he effec of performance sensiive fund flows on he efficiency of sock prices. They argue ha performance sensiive fund flows may lead o he failure of delegaed arbirageurs o exploi longerm arbirage opporuniies. When mispricing of an asse migh deepen before he arbirage opporuniy pays off, he arbirageur may be forced o unwind a posiion when i is leas profiable o do so. In anicipaion of his possibiliy, he arbirageur may no engage sufficienly in long-erm arbirage, and asses remain mispriced. In Shleifer and Vishny he inerim price risk associaed wih long-erm arbirage is due o he possibiliy of an inerim worsening of noise raders mispercepions concerning asse value. Thus, heir argumen is based on he assumpion ha shor-erm prices are inefficien and, in paricular, ha hey may be even more inefficien han he long-erm price. Alhough our model also exhibis he propery ha more inefficien shor-erm prices make long-erm arbirage less aracive, we find ha even wih perfecly informaive shor-erm prices, long-erm arbirage opporuniies may remain unexploied. Holden and Subrahmanyam (996) consider a model where he invesmen horizon is endogenous (long or shor) and raders are risk averse. They argue ha long-erm arbirage carries a higher risk, as more public informaion ges accumulaed ino he price of he asse. Risk averse agens may hus prefer shor-erm over long-erm arbirage, reducing he informaional efficiency of long-erm compared o shor-erm prices. 9 Despie exensive empirical research, he quesion wheher or no financial markes exhibi significan mispricing, remains unresolved. For an overview of he debae on marke efficiency see for example Fama (99) and Lo (997). 5
7. Conclusion In his paper we show ha incomplee informaion concerning he abiliy of a fund manager, may lead o shor-erm rade by he manager. This is he case, alhough wih complee informaion i is firs-bes o rade on long-erm informaion. If he invesor does no know he abiliy of he manager employed, she uses pas performance in order o learn abou he abiliy, and possibly o swich funds o anoher manager if his performance is bad. Our saring poin is he observaion ha ofen cied shor-run performance pressures ha may arise in his relaionship of delegaion, may no be a sufficien explanaion for possible shor-ermism, because even long-erm raders can consisenly earn profis in he shor-run, when hey are able o unwind heir posiions profiably afer a shor period of ime. Insead of focusing on possible incenive problems involved in implemening a paricular rading horizon, we explore he effec of differen rading horizons of he manager on he efficiency wih which he invesor can learn abou he manager s unknown ype. We show ha rading on shor-erm informaion allows more efficien learning abou he manager s abiliy for wo reasons. (i) A high abiliy manager can produce more precise informaion abou an even in he near fuure, compared o he more disan fuure, which means ha bad performance under long-erm rading is more likely o be aribuable o bad luck raher han low abiliy. As a resul, he performance observaions conain less informaion under long-erm rading. (ii) Under long-erm rading, informaion abou a manager s abiliy becomes available laer, because he principal can only evaluae he manager once he informaion on which rade occurred becomes public. This is he case even hough he manager can unwind he long-erm posiion profiably afer one period. The informaion conen of his immediaely available performance observaion depends on he informaional efficiency of shor-erm prices: he more informaive he prices concerning he shor-erm, he more informaive he immediaely available performance observaion. We show ha, even in he polar case, where shor-erm prices are perfecly informaive, (i) is sufficien o guaranee ha shor-erm rading is preferred by he principal for a non-empy se of parameer values. 6
APPENDIX A Proof of Lemma : When calculaing he expeced profis from rading on shor-erm informaion, noe ha he order for an asse eiher fully reveals he informed raders order (if Q k,+, {-n,n}) or does no reveal his order (if Q k,+, =0). Whenever he order is fully revealing, he marke maker ses he price of he asse equal o is expeced value, i.e. he asse is fairly priced. This means ha he manager does no make any profi in expecaion on such a rade. Generally, he shor-erm price of one asse depends on all variables in he marke makers informaion se (i.e. p A,+, depends no only on Q A,+, and w A,, bu also on Q B,+, and w B, ). However, when Q A,+, = 0, only w A, maers for p A,+,. To see why his is he case consider he following. In general he marke maker will use informaion obained in he marke for asse B +, o assess he likelihood ha a rader received a correc signal for asse A +. However, since he marke maker does no know in which direcion he informed rader raded in asse A +, no conclusions can be drawn, apar from wha he marke maker knows hrough receiving his own signal w A, for d A,+. As a resul, asses are priced independenly in all payoff relevan saes. Prices can be calculaed by Bayesian updaing and in he payoff relevan saes are simply p k,+, (Q k,+, = 0, w k, =)=ω p k,+, (Q k,+, = 0, w k, =0)=-ω Expeced wealh in + when rade on shor-erm informaion occurs a dae can be wrien as: E[W + a =a s, q ] = E[(W -θ A,+, p A,+, -θ B,+, p B,+, )(+r) + θ A,+, d A,+ +θ B,+, d B,+ a =a s, q ] = W (+r)+e[θ A,+, (d A,+ -p A,+, (+r))+θ B,+, (d B,+ -p B,+, (+r)) a =a s, q ] = W (+r)+ 8 ( ( ) ) ( ( ) ) ( ) ( ω ω [ q + + ] + ( )[ ( ( + ) + ) ( ν q qν ω ω q ν q qν) ] ) = W ( + r) + q νω( ω ) q.e.d. 7
Proof of Lemma : Noe ha when he marke maker ses prices a dae he has no privae informaion concerning he value of he asses A + and B +. Since asse values are independen and managers are ex ane equally likely o receive a wrong signal concerning eiher asse, he price for asse A + is independen of order flow in asse B + and vice versa. Using Bayesian updaing, prices can be calculaed as p k,+, (Q k,+, =, a l, q, w ) = /(+r) prob(d k,+ = Q k,+, =, a l, q, w ) ( ) = /(+r) ( ) prob Qk, +, = d k, + = prob d k, + = prob( Qk, +, = d k, + = ) prob( d k, + = ) + prob( Qk, +, = d k, + = 0) prob( d k, + = 0) ( ) ( ) ( ) µ ( ) ( ) = /(+r) γ µ + + γ µ µ γ ( µ + ) + ( γ ) + ( ) γ + γ = ( + r ) + Similarly he prices for Q k,+, = 0 and Q k,+, = - can be calculaed as p k,+, (Q k,+, = 0, a l, q, w ) = /(+r) / and p k,+, (Q k,+, = -, a l, q, w ) = ( + r ). In order o calculae he expeced rading profis of he long-erm rading sraegy, we need o know he price a which he marke maker is willing o unwind he manager s posiion. This price does no depend on he previous price p k,+,, because when he manager unwinds, he marke maker knows on which informaion he raded, since noise raders never unwind heir posiions premaurely. This renders he marke maker s previous belief (refleced in price) abou he manager s informaion irrelevan. The unwinding price, denoed by P k,+,+ (θ A,+,, θ B,+,, u +, w A,+, w B,+ ), hus depends on he orders θ k,+, submied a dae and he signals w k,+ received by he marke maker. The variable 8
u + denoes he manager s repuaion a he ime of unwinding. The repuaion u + may be differen from q and q +, because a dividend paymen d + occurs afer he order is originally submied and before he principal akes her employmen decision e +. Since he realisaion of d + may conain informaion abou he manager s abiliy, his repuaion a he ime of unwinding may be differen from q. The unwinding prices can be calculaed by Bayesian updaing: prob(d A,+ = Q A,+,, Q B,+,, w A,+, w B,+, u + ) = = x= 0 y= 0 x= 0 ( A, +,, B, +,, A, +, B, + +, A, + =, B, + = ) ( A, + =, B, + = ) prob Q Q w w u d d x prob d d x ( A, +,, B, +,, A, +, B, + +, A, + =, B, + = ) ( A, + =, B, + = ) prob Q Q w w u d y d x prob d y d x. When calculaing expeced rading profis under long-erm informaion acquisiion, only some of all possible unwinding prices are relevan. Recall he srucure of he model: dividend paymens are eiher zero or one wih equal probabiliy and noise raders eiher submi a buy or sell order of equal size and wih equal probabiliy. Moreover, all random variables are independen of one anoher and managers can eiher buy or shor-sell he asse. From his follows ha expeced rading profis are he same for any realised value of he dividend paymens d A,+ and d B,+. Hence, in order o calculae expeced rading profis i is sufficien o calculae possible asse prices and resuling rading profis for one paricular realisaion, say d A,+ = and d B,+ =. 9
The following able gives he relevan unwinding prices for he case d A,+ = and d B,+ = as a funcion of θ A,+,, θ B,+,, w A,+, and w B,+, w.l.o.g. for u + =γ. θ A,+, θ B,+, w A,+ w B,+ (+r)p A,+ P B,+ 0 prob () ω + ( ω ) ( )( ) ω + ω () ω () 0 (3) 0 (4) 0 0 ω (-ω) ( ω) + ω ( ) + + ( ) ( ) ω ω ω ω ω + ( ω) ( ) + + ( ) ( ) ω ω ω ω ( ω ) + ω ( ω ) + ω( ) (3) ω(-ω) () ω(-ω) (4) (-ω) (5) - ω ( ω) + ω ( ) + + ( ) ( ) ω ω ω ω (9) ω (6) - 0 ω + ( ω ) ( )( ) ω + ω () ω(-ω) (7) - 0 ( ω ) + ω ( ω ) + ω( ) (0) ω(-ω) (8) - 0 0 (-ω) ω + ( ω) ( ) + + ( ) ( ) ω ω ω ω () (-ω) (9) - ω ( ) ( ) + + ( ) ω ω ω ω (5) ω (0) - 0 ω ( ω ) + ω( ) (7) ω(-ω) () - 0 ( ω ) + ( )( ) ω ω (6) ω(-ω) () - 0 0 (-ω) ( ω ) ( ) + + ( ) ( ) ω ω ω ω (8) (-ω) 0 By symmery, he price for asse B + is he same as he price for asse A + in he corresponding row number. 30
The expecaion of wealh a dae + can be wrien as E [W + a =a l,q ] = E [(W - θ A,+, p A,+, - θ B,+, p B,+, )(+r) + θ A,+, P A,+,+ + θ A,+, P A,+,+ a =a l,q ] = W (+r) + E [θ A,+, (P A,+,+ -p A,+, (+r)) + θ B,+, (P B,+,+ -p B,+, (+r)) a =a l,q ] = W ( + r) + ( ) µ ( E u q ) + r + Noe ha since u + denoes repuaion before an employmen decision is aken, u + follows a maringale, i.e. E [u + ]=q. Hence, expeced wealh a dae + is W + r + q µ. + E [W + a =a l,q ] = ( ) ( ) r q.e.d. Proof of Proposiion : For q= a manager s repuaion never changes and hence i is opimal o induce he same sraegy in every period. The expeced discouned wealh from rading on long-erm informaion can be calculaed as (w.l.o.g. we consider opimisaion a dae =0) EW ( a ) = W0 + E l = ( + r) π a al q W = + µ, (5) 0 r( + r) ( = = ) Similarly he expeced discouned wealh under shor-erm rading is given by EW ( a ) = W0 + E s = ( + r) π = + ν, ω r (6) ( a = a q = ) W ( ω ) s 0 Seing EW (a l ) > EW (a s ) yields he inequaliy (9). From his i is also obvious ha (0) is a necessary condiion for ν>µ>µ *. q.e.d. 3
Proof of Proposiion : Since he principal receives a beer signal abou wheher or no he manager raded boh socks in he correc direcion, i is clear ha W * l (ω, η, u) is non-decreasing in ω. Noe ha for u=, he expeced payoff W * l (*) is independen of ω and η and given by equaion (5). In order o show ha he payoff o he principal is sricly increasing a some poin ω* [/,], we show ha for ω= he payoff is sricly greaer han for ω=/. For ω=, we can wrie W * l (ω=,η=0,γ) = [ π( l, γ ) + * l (,, ) + ( ) * l (, 0, γ )] + r E a W W Solving (above) for W l * (ω=,η=0,γ) yields W l * (ω=,η=0,γ) = * ( ) + W + r l (,, ) r + (7) (8) Now consider he payoff for ω=/. For ω=/ a newly employed manager who rades on long-erm informaion a dae will have an unchanged repuaion a + since no informaion concerning his abiliy becomes available (he unwinding prices are uninformaive if ω=/). This means ha he manager should no be fired afer he firs period. Therefore, W * l (ω=/,η=0,γ) mus saisfy W * l (/,0,γ) = /(+r){eπ(a l,γ) + W * l (/,,γ)}. (9) Now consider equaion (7) for W * l (/,,γ), which saisfies W * l (/,,γ) = /(+r){eπ(a l,γ)+w * l (/,,)+(-)max{w * l (/,0,γ),W * γ l (/,, ( µ ) )}}. We have o disinguish wo cases. Case : i may be opimal o fire he manager a + afer one bad performance, i.e. max{w * l (/,0,γ), W * γ l (/,, ( µ ) )}= W * l (/,0,γ). We can solve for W l * (/,0,γ) explicily. Sraighforward calculaion yields W * l ( /, 0, γ ) = * ( + r) ( ) + W ( /,, ) + r r + r + l (0) () Subsiuing W l * (/,,) given from (5) ino () and comparing (8) wih () yields W l * (,0,γ) > W l * (/,0,γ) >γ. We neglec W 0 for W l * (*), as i is irrelevan for he opimal decision. 3
Case : i may be opimal o reain he manager afer one bad performance, i.e. max{ W * l (/,0,γ), W * γ l (/,, ( µ ) )}= W * γ l (/,, ( µ ) Clearly, he value funcion is non-decreasing in repuaion q and since ( ) < γ W * l (/,,γ) > W * γ l (/,, ( µ ) γ µ Subsiuing his ino (0) yields ). ). W l * (/,,γ)(+r) Using (9), we can wrie ( + r ) + W l * (/,,) + (-) W * l (/,,γ). * W * l (/,0,γ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + W * + r l /,, + r + r + r + W /,, + l r + I is hen sraighforward o show ha for r< >γ W l * (,0,γ) > W l * (/,0,γ) q.e.d. APPENDIX B Proof of Theorem: The srucure of he proof is as follows. Firs, we show ha he condiion in () indeed is sufficien o make shor-erm rading opimal for he principal. Then we show ha here exis parameer values r, γ, µ, ν such ha for all ω>/, he sufficien condiion is saisfied and he parameers belong o he se F(ω) (Par (i) of he Theorem). Then we show ha he se of parameer values A(ω) is decreasing in ω (Par (ii) of Theorem). Proof of Par (i) Le V(η, u) denoe he highes possible expeced discouned profi from employing a manager who rades in he risky asse. We know from Proposiion ha when µ>µ *, in he firs-bes 33
case (u=) long-erm rading is induced and (5) gives he formula for he expeced discouned wealh in ha case. Moreover, i is clear ha when u= he value funcion does no depend on η, because he principal knows he manager s ype and learning ceases o be an issue. Hence, V(,)=V(0,) = µ r ( + r) Moreover denoe by V s (η, u) he payoff when a s is chosen in he nex period and he opimal decision aken ever hereafer. Similarly, define V l (η, u) as he payoff when a l is chosen. Thus, he value funcion can be wrien as V(η, u)= max{v s (η, u), V l (η, u)}. (3) Firs, suppose i is opimal o choose rade on shor-erm informaion for a manager of repuaion u. We can wrie ( ν ) u + [ ( )] u + ( ) = νω( ω) + ν ( ) + ( ν) V 0, u max q ( e ) q ( e ) V 0, q ( e ) V 0, s + r e where he disribuion of nex period repuaion u + is given by h(0,a s,q ): q qν u + = ( ν) wih q ν wih - q ν () ν (4) (5) If u =γ, clearly i is beer o fire he manager afer one bad performance han o reain him, since a manager from he pool has a higher repuaion. Hence, if a s is indeed opimal a u =γ, we can wrie ( ) u V( 0 u ) ν +, ν + = V s (0, γ). Subsiuing his ino (5), we can solve for V s (0,γ). V (, γ ) s 0 = ( ) + V (, ) γνω ω γν 0 r + γν (6) Now suppose i is opimal o induce rade on long-erm informaion for a newly employed manager. To sar wih, consider he case where ω=. As was shown in he proof of Proposiion (and in he ex of secion 5), a manager opimally ges fired afer one period if performance is bad. The repuaion of a manager employed can herefore only ever ake one of wo values: eiher q=, if he manager raded boh asses in he correc direcion, or q=γ, if rade in one asse was in he wrong 34
direcion and a new manager ges employed. Moreover, if a u =γ, i is indeed opimal o induce rade on long-erm informaion, i will be opimal o do so, for any u τ in he fuure, since u τ {γ, }. Hence, for ω=, W * l (ω=,η, u=γ)=v l (η, u=γ). Suppose now ha he principal is able o observe wheher or no he choice of rades by a long-erm manager was correc afer one period, even when ω<. This makes he payoff from inducing rade on long-erm informaion a leas as high han payoff from he acual iming of performance observaion prevalen in he model. Thus, W * l (ω=, η, q=γ) consiues an upper bound for he payoff ha can be achieved by inducing rade on long-erm informaion wih a newly employed manager and following an opimal employmen policy. Therefore, W * l (ω=, η=0, γ) V l (η=0, γ). (7) From (8), we know ha W l * (ω=, η=0, γ) = * ( ) + W + r l (,, ) r +. Moreover, a u=, obviously, W * l (ω, η, u=) = V(η,) µ>µ *. Subsiuing his ino (8) allows us o rewrie (7) as ( ) V l (0,γ) ( ) + V, + r r + (8) Comparing (6) and (8) yields V s (γ) > V l (γ) rγv(,) (ν-µ) > γ µ r νω( ω ) + γ µν ω ( ω ). (9) ( + r) ( + r) Subsiuing () ino (9) and rearranging he resuling inequaliy yields he inequaliy in (). Moreover, seing µ, ν, r, ω such ha µ=µ * +ε, wih ε>0, we ge ν µ r + γν > ε µ * γ + ε ( ) (30) For µ * >0, he RHS of (30) ends o zero as ε 0, and hence for any ν>µ an ε > 0 can be found such ha (30) is saisfied. If µ * =0, he RHS of (30) is equal o. Again, parameers can be found such ha (30) is saisfied. An example is r=0.0, γ=0.5, µ=0.9, ν=0.6. Noe ha in he case where µ * =0, any se of parameers ha saisfies (30) implies </4, which is why no parameer resricion 35
could be made on. However, he resul is no driven by he fac ha a manager may deviae from following his signal under rade on long-erm informaion (which could only reduce he invesor's payoff from inducing rade on long-erm informaion). Proof of Par (ii) In order o prove saemen (ii) of he Theorem, we will show ha he principal s payoff is decreasing in ω when shor-erm rading is opimal and ha i is non-decreasing in ω when longerm rading is opimal. Suppose ha a (η =0, q =γ), a =a s is opimal. From (6) i can be seen ha he efficiency parameer ω only eners V s (0,γ) hrough he insananeous payoff. I is hen sraighforward o show ha for ω (/, ], V s 0 (, γ ) ω V < 0. Only for ω=/, we ge s 0 ω (, γ ) = 0. Hence, if a s is opimal a (η =0, q =γ), he payoff V(0,γ) is decreasing in ω. Now suppose ha a (η =0, q =γ), a =a l is opimal. For rading under long-erm informaion, he efficiency of shor-erm prices does no affec he insananeous profi (see equaion (8)). Insead ω only affecs he belief updae of he principal. In paricular, an increase in ω leads o more efficien learning abou he manager, because he price P k,+,+ is more informaive abou he rue sae of he world (d k,+ ) and hus abou wheher or no he manager received a correc signal. Since he principal uses his informaion opimally, he expeced payoff mus be non-decreasing in ω when V(0,γ) = V l (0, γ). From his i follows sraighforwardly ha an increase in ω reduces he principal s incenive o induce shor-erm rading, compared o long-erm rading. Therefore, if for a given vecor x = (r µ ν γ) shor-erm rading is induced a ω i will also be induced a ω <ω. On he oher hand, here always is a vecor x = (r µ ν γ) ha lies sufficienly close o he boundary of A(ω ), ha i will no lie in A(ω ) for all ω >ω. Hence, A(ω ) A(ω ) for ω >ω. q.e.d. 36
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