Pric Co-Movmnts and nvstmnt Funds Maryam Sami Dpartmnt of Economics, Stony Brook nivrsity Job Markt Papr Abstract This papr discusss pric co-movmnts btwn fundamntally indpndnt financial markts populatd by risk nutral global funds and spcializd funds. Th invstmnt dcisions ar dlgatd to risk nutral fund managrs who ar informd or uninformd of th stat of th markts and hav rputational concrns. W show that in any quilibrium of th modl, prics of th risky assts co-mov with ach othr following any shock to x-ant probabilitis of dfault. Th mchanism that gnrats this co-movmnt rlis on two sourcs: th information asymmtry btwn fund managrs and th rputational concrns of uninformd fund managrs facing th thrat of dismissal. Th rputational channl rinforcs th co-movmnt but it is not ncssary to gnrat it. nformation asymmtry inducs co-movmnt vn in th absnc of rputational concrns. Kywords: Pric Co-Movmnt, Global Funds, Spcializd Funds. JEL classification: G, G2. ntroduction According to Nw York Stock Exchang Factbook, in 2003 institutional invstors hld almost 50% of corporat quity in NYSE. n 950, this numbr was only 7%. nvstors rward fund managrs according to som masur of thir succss in gnrating rturns and withdraw thir funds if thy dm maryam.sami@stonybrook.du
th managr incomptnt and unsuccssful. So th managrs incntivs ar two fold; thy want to maximiz th rturn on thir portfolio and build up a rputation for thmslvs as comptnt managrs. Thr is a growing litratur on th gnral quilibrium implications of institutional trading that discusss th pric distortions gnratd by th incntivs of fund managrs. Alongsid th shift from individual invstors to institutional invstors, thr hav bn pisods of th sprad of financial crisis btwn mrging markts that had no common fundamntals. A good xampl is th 998 ussian Flu that sprad to Brazil. Th common notion in th litratur about ths pisods has bn multiplicity of quilibrium du to financial vulnrability and markt incompltnss. ntrstingly, th affctd markts wr all populatd by institutional invstors such as global hdg funds. At th sam tim, thr has also bn a ris in intrdpndnc and comovmnt btwn stock prics all ovr th world that is not xplaind by common fundamntals, global shocks and changs in volatility. 2 Our aim is to addrss th quilibrium consquncs of having spcializd and global invstmnt funds, dlgating th invstmnt dcision to rputationally motivatd managrs for pric co-movmnt btwn fundamntally indpndnt markts. Our ky assumption is th asymmtric information among managrs. Managrs can b informd of th tru stat of th assts or uninformd. W show that in any quilibrium of th modl, prics co-mov with ach othr following any shock to th prior blifs about th markts. Our modl builds on Sami and Brusco (204) and Gurriri and Kondor (202). Thr ar two fundamntally indpndnt risky assts and a risk lss bond. W hav thr typs of funds; spcializd in markt on, spcializd in markt two, and global. isk nutral fund managrs ar ithr informd or uninformd and ar hird to invst th mony of risk nutral invstors. Typs of funds ar obsrvabl but typs of managrs ar privat information. Also, thr ar indpndnt masss of liquidity tradrs at ach risky asst markt. Managrs ar paid a fixd shar of rturn thy hav mad and ar rtaind by th funds if thy hav mad th highst possibl rturn fasibl for thm. This mans that spcializd funds rtain th managr if h has bought th rpaying asst or th risk-lss bond whn th asst For xampl, Cuoco and Kanil (2006), Dasgupta and Prat (2008), Basak and Pavlova (20) 2 Forbs (202) survys mpirical and thortical litratur on contagion and documnts significant ris in co-movmnt btwn stocks within advancd countris, Euro rgion and all ovr th world controlling for global shocks and changs in volatility. Anton and Polk (204) idntifis a significant incras in th rturn co-movmnt of th stocks hld by th sam mutual funds. 2
dfaults. Howvr, global funds rtain thir managrs if thy hav bought th rpaying risky asst with th lowst pric or risk fr bond whn both assts dfault. As in Sami and Brusco (204), w considr partially rvaling rational xpctation quilibria. W focus on quilibria in which if p is an quilibrium pric at a crtain valu of th liquidity and rturn shock ralization, and at p th xcss dmand is idntical for anothr shock ralization, thn th quilibrium pric must b th sam. f asst i rpays, p i rvals it s rpaying whn it is qual to, whr is th rturn on risklss bond with a pric normalizd to. f asst i dfaults, p i rvals th dfault if it is lss than or qual to p i which clars th markt with th dmands of liquidity tradrs 3. n Sami and Brusco (204), w showd that thr is no quilibrium at which prics don t rval any information about th tru stat of th assts. Bsids, whn all th funds ar global, thr is no partially rvaling quilibrium with diffrnt unrvaling prics, i.., in any quilibrium, unrvaling prics must b qual. n this papr, first prov that as long as thr ar global funds in th markt, prics ar intrdpndnt in any quilibrium. Consquntly, intrdpndnt prics comov with ach othr following any shock to th priors on th assts. Th rsult is obtaind dspit th fact that all agnts ar risk-nutral. This comovmnt is magnifid by rputational concrns of managrs but dos not go away if thr is no rputational concrn. Morovr, w show that whn thr ar htrognous funds, w hav both typs of quilibria, quilibria with qual and unqual unrvaling prics. Th analysis in Sami and Brusco (204) was only limitd to on typ of quilibria-simpl quilibria- whil in this papr w charactriz both simpl and non-simpl quilibria. 4 Th mchanism that gnrats th intrdpndnc and co-movmnt rlis on two sourcs, th information asymmtry btwn fund managrs and th rputational concrns of uninformd fund managrs facing th thrat of dismissal by funds. nformd managrs ar prfctly informd and hav strict dmands for th rpaying asst. also assum that thr ar vry fw informd managrs and a lot of uninformd managrs in th markt so that th dmands of informd manags can t clar th markt so uninformd managrs must hav positiv dmand for th assts for th markt to gt clard. Now, imagin that th risky assts ar ussian bond and Brazilian bond. Suppos that ussian bond dfaults and Brazilian bond rpays. Global informd managrs know this and thy all dmand Brazilian 3 W will latr on solv for p i in quilibrium. 4 An quilibrium is simpl if only on unrvaling pric vctor occurs in quilibrium. 3
bond. But thn th probability that uninformd managrs rciv Brazilian bond is lss than rciving ussian bond simply bcaus all th informd managrs dmand Brazilian bond. This shows a coupl of things. First uniformd managrs fac an advrs slction problm, with highr probability thy rciv th dfaulting bond(ussian bond). Scond, th probability of rciving Brazilian bond dpnds on th stat of th ussian bond and dcrass with th incras in th dfault probability of ussian bond. So uninformd managrs dmand th bonds if prics compnsat thm for this advrs slction problm. Also, prics must co-mov with ach othr following any chang in x-ant dfault probability of on of th bonds. To s this, suppos that th x-ant dfault probability of th ussian bond incrass. Clarly, pric of ussian bond suffrs. But at th sam tim, uninformd managrs would rationally bliv that if Brazilian bond has rpaid, th probability of rciving it is now vn lss. So to compnsat th uninformd for th ris in th risk of not rciving th rpaying Brazilian bond, th pric of Brazilian bond must also go down. Sinc assum that th total mass of informd managrs and liquidity tradrs is nvr nough to clar th markts, unrvaling prics ar claring th markts only if thr is a positiv dmand from uninformd managrs. ninformd managrs hav positiv dmand for risky assts if prics compnsat thm for th risk of bing dismissd. This mans that thy ask for prmia ovr th rturn of th risk fr bond which ar not indpndnt of ach othr. This prmia incrass th pric co-movmnts, howvr, th co-movmnt dosn t disappar if thr is no rputational concrn and th prmium is zro. n othr words, if instad of dlgating th invstmnt, invstors dirctly invst in th markts, uninformd tradrs fac th sam signal xtraction problm of uninformd managrs. Thy hav to larn th signals of informd tradrs from prics. Sinc prics rflct th signals of informd global tradrs, thy ar intrdpndnt and co-mov with ach othr following any chang in th x-ant probabilitis of th dfault of any asst. Litratur viw. This papr is an xtnsion of Gurriri and Kondor (202). n a modl with only on risky asst, on risk fr bond and on typ of invstors, thy show that th rputationally concrnd managrs distort th pric of th risky asst by asking a prmium ovr th risk fr bond that compnsats thm for th risk of gtting fird and maks th pric mor volatil. Our papr contributs to th litratur on gnral quilibrium modls of contagion with information asymmtry and dlgation. Th closst modls to ours ar th modls that discuss information channls of contagion and th contagion du to dlgation. Calvo (999) has a rational xpctation modl at which uninformd tradrs s th actions of informd 4
tradrs but fac a signal xtraction problm; whn informd tradrs don t buy an asst, uninformd tradrs don t know if this is bcaus of a ngativ idiosyncratic shock to thir dmand or it is bcaus of a ngativ shock to th valuation of th assts. Thus, whn th volatility of th rturns in mrging markts ar rlativly highr than th volatility of th idiosyncratic shocks, following a ngativ shock to on markt uninformd tradrs attach highr probability to th low rturn for othr markt as wll. Th main diffrnc btwn Calvo (999) and us is th pricing mchanism; at his modl uniformd tradrs first obsrv th actions of informd ons and thn choos to buy or sll mrging markts. n our modl, all th tradrs mov simultanously and it s only th pric that rvals information to th markt. Chakravorti and Lall (2005) hav a gnral quilibrium modl of dlgatd portfolio managmnt. Thy hav ddicatd and opportunist managrs. Ddicatd managrs only invst in mrging markts and ar compnsatd basd on th xcss rturn that thy mak ovr a bnchmark indx of mrging markts. Opportunist managrs ar allowd to short sll and ar payd a fixd shar of th total rturn mad on th portfolio. Thy show that pric co-movmnt btwn mrging markts is th rsult of th portfolio r-balancing by managrs following a shock to on markt. Our modl diffrs from thm in having asymmtric information as th main sourc of pric co-movmnt. Dasgupta and Prat (2008) is a squntial trading modl with on risky asst that xtnds Glostn and Milgrom (985) modl by introducing carr concrnd tradrs. Thy show that managrs with rputational concrns distort th pric so that it nvr rvals th tru stat of th asst. Kodrs and Pritskr (2002) has a rational xpctations modl of contagion with asymmtric information whr fundamntally unrlatd markts can xprinc contagion du to th cross-markt r-balancing. Thr is no contagion in thir modl whn fundamntals and liquidity shocks ar uncorrlatd. Finally, our modl is rlatd to th big litratur on contagion du to hrding. n Scharfstin and Stin (990) managrs follow ach othr to avoid bing rgardd dumb and shar th blam if th things go wrong. n a mor rcnt papr, Wagnr (202) shows that th thrat of dismissal by invstors inducs th managrs to fir sals and run whn thy suspct othrs would do th sam to avoid slling th assts at lowr prics latr vnif thy ar not going to b valuatd in th futur. Th rst of this papr is organizd as follows. Nxt sction prsnts th modl. n sction 3, w charactriz th quilibrium. Sction 4 contains concluding rmarks. 5
2 Modl Thr ar two risky assts and on risk fr bond paying >. Th rturn on risky asst i at tim t is dtrmind by th ralization of a random variabl χx i,t which taks valus in th st {0, }. Th ralization of χx t = (χx,t, χx 2,t ) is dnotd χ t = (χ,t, χ 2,t ). f χ i,t = 0 thn th asst rpays an amount of, whil if χ i,t = th asst dfaults and pays zro. Th random variabls {χx i,t } t=0 ar all indpndnt and idntically distributd, with Pr (χx i,t = ) = q i and q 2 > q. Furthrmor, ach χx i,t is indpndnt of all variabls {χx j,τ } with j = i. τ =0 isky assts ar sold at prics p i. Thy ar supplid in fixd inlastic amounts of b and b 2. Lt b = (b, b 2 ) b th vctor of supply. Thr is also a prfctly lastic supply of risk fr bonds at pric. W hav thr kinds of agnts; invstors, fund managrs and liquidity tradrs. nvstors ar ndowd with on unit of capital but thy can t invst it thmslvs and hav to hir fund managrs. nvstors ar of thr typs, only invsting in asst and bond,, invsting in asst 2 and bond, 2, or invsting in both assts and bond, 3. W assum that th mass of j invstors is also j. W can think of ach typ of invstor as a typ of fund. Fund managrs ar also of two typs; informd () and uninformd (). nformd managrs obsrv th ralizations of χx i,t for i =, 2. ninformd managrs only obsrv prics of th assts. Th typs of invstors ar obsrvabl whil th typs of managrs ar privat information. Th mass of informd managrs(m ) is lss than th mass of any fund j. Liquidity tradrs ar only dmanding risky assts for random rasons. Lt y and y 2 b th masss of liquidity tradrs at ach asst markt. W assum that y i s ar indpndntly and idntically distributd according to th uniform distribution ovr [y, y]. At th bginning of ach day, funds with no managr ar randomly matchd with a managr in th unmploymnt pool. W assum that funds looking for a managr ar not obsrving th prvious history of any mploymnt of th managrs in th unmploymnt pool. Funds offr th matchd managr a contract that pays a fixd shar of rturn γ, and rtains him only if th managr has mad th highst possibl rturn. W will discuss th asst and labor markts in dtail latr but bfor that w prsnt th tim lin of th modl. 2. Timing Th timlin of th modl is as follows; 6
n th morning nmployd managrs dcid to pay th sarch cost κ and ntr th unmploymnt pool or stay out of markt. Funds with no managr randomly pick a fund managr from unmploymnt pool. nformd managrs obsrv th ralization of rturn shocks χ t. Managrs choos thir dmand of th assts and th bond. Equilibrium prics p t = (p t, p 2t ) ar dtrmind and th assts ar allocatd. n th vning, χ t is publicly obsrvd and th invstmnts of th managrs ar ralizd by thir invstors. Managrs rciv a shar γ of th rturns. Any fund rcivs an xognous binary signal, σ l, about th typ t of managr l. f th managr is informd, thn σ l is always zro. t Othrwis, σt l = 0 with probability ω and σt l = with probability ω. Funds dcid to fir or rtain thir managrs. With probability δ any managr is xognously sparatd from th job. 2.2 Labor Markt To hir a managr ach fund randomly picks a managr from th pool of unmployd managrs Lt Z t = Zt + Zt b th total mass of unmployd managrs of both typs and A t th mass of funds looking for a managr at any tim t. Also dfin µ t as th probability of matching. Sinc funds and managrs ar matchd randomly th probability that a managr is matchd is: min{a t, Z t } µ t = () Z t Clarly, funds dcision to fir or rtain any managr aftr obsrving managrs rturns dpnds on th matching probability µ t, th fraction of in- Zt formd unmployd managrs out of all unmployd managrs Z, and thir t updatd probability about th managrs comptnc. Lt N i j b th st of 7
th managrs of typ i =, hird by th funds of typ j. Lt also φ j (θ q j, σ q, p j, χ j ) {0, } dnot th rtntion dcision of fund j aftr ob q srving th invstmnt dcision θ j of th managr q, th xognous sparation signal σ q, quilibrium pric(s) and th tru valu of th asst(s) χ j. By th sam argumnt, lt φ 3 (θ q 3, σ q, p, χ) {0, } b th firing dcision for 3 funds. Thn, φ j = 0 if th managr is rtaind and φ j =, othrwis. Th invstmnts of managrs in ( 2 ) funds ar succssful if thy buy risky asst (2) whn it rpays and buy risk fr bond whn it dfaults. For managrs in 3 funds, th invstmnt is succssful whnvr thy buy risk fr bond whn both assts dfault or thy buy th chapst asst that is rpaying. 2.3 Asst Markts Each managr submits a dmand schdul. Managrs in and 2 funds can dmand risk fr bond, th risky asst th fund spcializs in, or stat indiffrnc btwn thm. Managrs hird by 3 funds can dmand ach of th risky assts, risk fr bond or b indiffrnt for a subst of assts. Th auctionr collcts th dmand schduls, sts th markt claring prics and allocats th assts to managrs and liquidity tradrs. Givn th submittd dmands of managrs, th auctionr first assigns th managrs with th strict dmand of asst, asst 2 or risk fr bond and thn assigns to th managrs stating indiffrnc btwn th invstmnt opportunitis at prics that clar markts. N and N2 managrs submit th dmand schduls d j(p j χ j ) : [0, ] {0, } {0, } 2, j =, 2 to th auctionr. f d = (0, ) for som χ and p, thn th managr dmands no bond and /p units of risky asst whil d = (, ) mans that th managr is indiffrnt btwn unit of bond or /p units of risky asst. Givn χ = (χ, χ 2 ) and p = (p, p 2 ), N3 managrs submit d 3 (p χ) : [0, ]2 {0, } 2 {0, } 3 to th auctionr. Finally, uninformd managrs hird at j funds, N j, hav no privat signal so whn hird by or 2 funds, thir dmand schduls ar givn by d j (p j ) : [0, ] {0, } 3 whr d = 0 for k / {0, j}. f hird by 3 funds, uninformd jk managrs dmand is givn by d 3 (p, p 2 ) : [0, ] 2 {0, } 3. Lik managrs, liquidity tradrs ar also ndowd on unit of capital that thy invst it ntirly on a risky asst. At any pric p it liquidity tradrs in markt i buy /p i units of asst i. Throughout th papr, w assum that b i > y so thr is always sufficint supply to covr th dmands of liquidity tradrs. Now, assum that asst i is xpctd to dfault and th only agnts that 8
still dmand th auctionr to assign thm asst i ar liquidity tradrs. Th auctionr clars th markt by assigning th ntir b i units of asst i to y it y liquidity tradrs at p it (y it ) =. Not that p it (y it ) [ y, ]. This mans b i b i b i y that in quilibrium, any pric blow bi automatically rvals that th asst y is dfaulting. From this point on, lt p i = b i and p = max{p, p 2 }. Dfin W j, j =, 2, 3, as th continuation payoff for an uniformd managr of bing mployd at fund j. Also dfin v j (k, p j ) as th xpctd payoff of N j managr, j =, 2, buying asst k = 0, j. W hav, v j (k, p) = E[γ j + ( φ j (θ q, σ q, p j, χ j ))βw j p = (p, p 2 )] (2) j whr if k = 0 j = if k = j χj p j Now lt v 3 (k, p, p 2 ) b th payoff of N3 i managr buying asst k = 0,, 2. Thn, whr, v 3 (k, p, p 2 ) = E[γ 3 + ( φ 3 (θ q, σ q, p, χ))w 3 p = (p, p 2 )] (3) j if k = 0 χ 3 = p if k = χ 2 p 2 if k = 2 Th payoffs for informd managrs ar dfind th sam as (2) and (3) but not that N and N2 managrs rciv singl prfct signals χ j and thir payoff is v j (p j χ j ). N3 managrs obsrv χ = (χ, χ 2 ) and hnc thir payoff is dnotd by v 3 (p χ) and is givn by v 3 (k, p, χ)) = E(γr 3 + ( φ j (θ, σ, p, χ))βw χ) (4) 3 Dfin th st of all possibl dmand vctors as 2 Δ = {(d 0, d, d 2 ) d i } (5) Lt A(p) = A (p), A 2 (p) b th aggrgatd dmand vctor at pric p, that is A(p) = d q (p)dq (6) q N i=0 9
whr N is th st of all tradrs. Lt x k (d q ; A k ) : Δ [0, ] whr 2 k=0 x kd q = dnots th fasibl allocation to a managr with dmand d q. W hav now dfind all th lmnts of th quilibrium and ar rady to dfin th quilibrium. Dfinition. Givn any collction of (Nj, Nj, W j ), j =, 2, 3, th ra tional xpctations quilibrium consists an quilibrium pric mapping p : y y {0, } 2 [y, y] 2 [ b, ] [, ]; quilibrium dmand schduls d i j, for b 2 i =, and j =, 2, 3; and fasibl allocation mapping x k (d i j ) [0, ], for ach asst k such that, (A). th pric vctor p(χ, y) = (p (χ, y), p 2 (χ, y)) clars th markts. That is, for asst k =, 2, q N k whr dˆq = d j i (p,.). x k (dˆq )dˆq dq+ x k (dˆq )dˆq dq + x k (dˆq )dˆq dq k k k q N3 q Nk + x k (dˆq )dˆq dq = p k b k y k (7) q N 3 (A2). th dmand schduls of N j managrs ar optimal givn p(χ, y).that is, if d i = thn v (k, p) v (k ', p) for all k ' = k. kj j j (A3). th dmand schduls of N j, j =, 2, and N managrs ar optimal 3 givn p(χ, y), i.., d j (p j χ j ) = and d j 3 (p χ) = for χ j = 0, and d j (p j χ j ) = 0 and d j3 (p j χ j ) = 0 for χ j = ; j =, 2. Lt D k (p) b th st of all th tradrs with strict dmands for asst k at pric p. That is, for q D k (p), d q k = and dq j = 0 for j = k. Lt Z k(χ, y) b th mass of all d q, q D k (p) at (χ, y) That is, d q q D k (p) k Z k (χ, y) = kdq (8) Th quilibrium that w construct satisfis th following blif consistncy condition. Dfinition 2. Lt p b a rational xpctations quilibrium pric mapping and p (χ, y) b th quilibrium pric vctor at (χ, y). Also assum that thr xists (χ ', y ' ) such that Z(χ, y) = Z(χ ', y ' ). Thn, p is blif consistnt if p (χ, y) = p (χ ', y ' ). 0
Dfinition 2 rstricts th st of quilibria to th partially rvaling quilibria. f th quilibrium pric vctor is blif consistnt thr ar som valus of y that pric vctor is not rvaling χ. Hnc, th quilibrium pric vctor that always rvals th rpay or dfault of th assts is not blif consistnt. Bfor moving on to th nxt sction, w introduc anothr fatur of our quilibrium. Dfinition 3. An quilibrium pric mapping p (χ, y) is simpl if thr is at most on pair (p, p 2 ) with p i (p, ), i =, 2, such that p (χ, y) = (p, p 2 ). An quilibrium is non-simpl if th quilibrium pric mapping p (χ, y) taks mor than on valu in (p, ) 2. 3 Equilibrium W construct a class of stationary quilibria at which N i = N i jt j, µ jt = µ j, and W = W. n non of ths quilibria prics ar fully rvaling, so funds jt j hav highr xpctd payoff if thy hav an informd managr. This suggsts that rputation is valuabl for uninformd managrs as wll, bcaus any mistak lads to dismissal and loss of W j. Hnc, from this point on w tak ({N j, N j, W j, µ j } j=,2,3 ) as givn and discuss th xistnc and proprtis of th rational xpctations quilibrium at asst markts. W also assum that it is optimal for th funds to fir any managr who hasn t mad th highst possibl rturn. Latr on w solv for quilibrium N j i and W and j prov th optimality of th firing rul. Th xistnc of this class of quilibria is guarantd undr th following assumptions; max{b, b 2 } M < min{y, y y}, M + y < C, < y + min{, 2, 3 } (9) whr C is givn in th Appndix. Th first part nsurs that th mass of informd managrs is small rlativ to nois tradrs, making th quilibrium not always fully rvaling. Th scond part nsurs that th total invstmnts of informd managrs and nois tradrs ar nvr nough to clar th markts so thr is always som amount of ach asst that is allocatd to uninformd managrs. Howvr, by th third part th supply is nvr nough to allocat risky assts to all uninformd managrs. Thus, in quilibrium uninformd managrs ar always indiffrnt btwn risky asst(s) and riskfr bond.
ω > +δ This assumption is idntical to th assumption mad by Gurriri and Kondor (202). Aftr obsrving a right dcision by a fund managr at th nd of ach day, funds attach a highr probability to th vnt that th managr is informd than uninformd. This assumption nsurs that in quilibrium, th blifs of funds about succssful managrs grows at th high nough spd so that thy ar rtaind aftr a right dcision. κ < γ This assumption nsurs that th sarch cost is not mor than th xpctd payoff of gtting hird for uninformd managrs. t xcluds quilibria which all unmployd uninformd managrs ar matchd with probability. Givn ths assumptions, w discuss how information sprads into th markt from th dmands of informd managrs. Bfor charactrizing th quilibria w nd to introduc th concpt of marginal tradrs that is going to play a cntral rol in th charactrization of ach quilibrium. 3. Marginal Tradrs Dfinition 4. Suppos p (χ, y) is an quilibrium pric mapping with p (χ, y) = (p, p 2 ) (p, ) 2 for som (χ, y). Also assum f j = P r(χ j = p = (p, p 2 )). Thn, N j managrs j =, 2, 3 ar marginal tradrs at p = (p, p 2 ) if v j (j, p) = v j (0, p) By Dfinition 4, if N j j =, 2, 3, ar marginal tradrs at p = (p, p 2 ), thir xpctd payoff of buying asst j is qual to th xpctd payoff of buying risk fr bond. This mans that N j ar indiffrnt btwn asst j and risk fr bond at p j. This also implis that p j is th maximum pric that marginal tradrs ar willing to pay for asst j. At any pric abov p j thy nvr dmand asst j. writing th condition in Dfinition 4 for j =, 2, w hav γ ( f j )( + δωβw j ) = γ + δωf j βw (0) p j Th right hand sid of (0) is th xpctd payoff of buying asst j. call that uniformd managrs in spcializd funds ar paid γ shar of th rturn and ar only rtaind if thy buy risky asst whn it rpays and risklss bond whn risky asst dfaults. Thus, thir xpctd payoff of buying asst j 2
γ( f j ) j is th xpctd rtrun on asst j, p j, plus th xpctd payoff of bing rtaind. But th probability of bing rtaind for N j whn buying asst j is th probability of th rpay of asst j, f j, tims th probability that h is not xognously sparatd, δ, tims th probability that his typ is not rvald ω. Th lft hand sid of (0) is th xpctd payoff of buying th saf asst. Now, th managr buying risklss assst is rtaind only if risky asst j has dfaultd, hnc th xpctd payoff of bing rtaind is δωf j βw j. Now suppos N3 managrs ar marginal tradrs at (p, p 2 ) and p > p 2. This mans that ( f )( γ + δωf 2 βw 3 ) = γ + δωf f 2 βw () 3 p ( f 2 )( γ + δωβw 3 ) = γ + δωf f 2 βw (2) 3 p 2 Not that whn p > p 2, N managrs buying asst ar only rtaind whn 3 asst rpays and asst 2 dfaults, bcaus if asst 2 rpays, th rturn on asst 2 is highr than th rturn on asst and N ar only rtaind whn 3 thy buy th asst that pays th highst rturn. Thrfor, th probability of th rtainmnt for a managr buying asst is δω( f )f 2. Howvr, th managr that buys asst 2 is rtaind whnvr this asst rpays irrspctiv of th dfault or rpay of asst and his probability of rtainmnt is qual to δω( f 2 ). Any N managr who buys risk fr bond is only rtaind 3 if both assts dfault. This mans that th probability of th rtainmnt is δωf f 2. Not that if p = p 2, both assts ar paying th sam rturn if thy rpay. Thus, th indiffrnc conditions for N managrs ar 3 γ ( f j )( + δωβw 3 ) = γ + δωf f 2 βw (3) p j whr j =, 2. Not that, N3 managrs continuation payoff of bing mployd, W 3, is lss than th continuation payoff of uninformd managrs of spcializd funds. This is bcaus if both risky assts rpay, N3 managrs who buy th mor xpnsiv asst ar fird. Nvrthlss, N j,j =, 2, managrs buying th rpaying risky asst ar always rtaind. Notic that if W 3 W j, γ + δωf j βw j > γ + δ + ωf f 2 βw (4) 3 f p = (p, p 2 ), p > p 2, occurs in quilibrium and marginal tradrs ar N j managrs, (4) and indiffrnc conditions ()-(2) imply that at p = 3 3
(p, p 2 ) th payoff to N3 managrs of buying risklss bond is lss than th payoff of buying asst j. Whn Nj ar marginal tradrs at p = (p, p 2 ) th maximum pric that N3 ar willing to pay for asst j is always highr than p j. Thrfor N 3 managrs ar not marginal tradrs at p = (p, p 2 ) and strictly dmand th chapst risky asst. Lt P jj and P j3 dnot th maximum prics that Nj and N3 pay for asst j. This mans that P jj and P j3 ar solvd from (0) and ()-(2) and ar givn as γ( f j ) P jj = (5) γ + (2f j )δωβw γ( f j ) P j3 = (6) γ + (f f 2 f j )δωβw 3.2 nformation vlation in Asst Markt Equilibrium n quilibrium, pric is a mapping from th spac of stochastic shocks (χ, χ 2 ) (y, y 2 ) to th intrval [ y, ] [ y, ]. Clarly, th invrs mapping b b 2 y y b b 2 (p ) (p) at any p [, ] [, ] is a subst of {0, } 2 [y, y] 2. So any p = p (χ, y) is in principl rvaling information about (χ, y). Now th qustion is, how much information is rvald at a blif consistnt quilibrium? s thr any blif consistnt quilibrium at which pric dos not rval any information, that is p (χ, y) = p for all (χ, y)? s thr any quilibrium that is fully rvaling, i., p j(χ j, y) {, p } for any (χ j, y j )? s thr any j quilibrium that is rvaling for som (χ, y) and unrvaling for othr valus of (χ, y)? Bfor answring ths qustions, lt us first stat th following proposition about th proprtis of th quilibrium pric mappings. Proposition. Any rational xpctations quilibrium pric mapping p (χ, y) satisfis th following conditions; (i). f p =, thn χ i = 0. i y (ii). f p [, y ], thn χ i =. i b i b i Th proof of th abov rsult is vry simpl and is omittd. f thr is any quilibrium at which p i = whn χ i =, thr would b no dmand from informd or uninformd managrs to buy asst i and only nois tradrs dmand asst i at But thn to clar th markt p must b y i and not.. i b i Proposition 2. ndr assumption (9), 4 j 3
(A). Thr is no blif consistnt unrvaling quilibrium. max{b,b 2 } (A2). Suppos < y + min{, 2 }. Thr xists a rvaling quilibrium. This quilibrium is not blif consistnt. Ths rsults ar similar to th rsults in Sami and Brusco (204) and thir proof is prsntd in th on-lin appndix of th papr. W know by th abov proposition that non of th two xtrms, no rvlation and full rvlation ar possibl or plausibl. So th quilibrium must always b partially rvaling; somthing is always lakd to th markt. ndd, this is th cas in th bas lin modl of Gurriri and Kondor (202). W show that thr xist simpl and non-simpl partially rvaling quilibria with a common proprty; whn prics ar not fully rvaling thy ar intrdpndnt. Dfinition 5. Suppos p (χ, y) = (p (χ, y), p 2 (χ, y)) is an quilibrium pric mapping. Thn p (χ, y) and p 2 (χ, y) ar intrdpndnt if thr is at last on pair (p, p 2 ) (p, ) 2 ; p i (χ, y) = p i for som (χ, y) such that P r(p = p, p = p 2 χ, χ 2 ) = P r(p = p χ )P r(p = p 2 χ 2 ). 2 2 W call p (χ, y) and p 2 (χ, y) indpndnt if thy arn t intrdpndnt. To undrstand Dfinition 5, suppos p (χ, y) is a simpl quilibrium pric function whr p (χ, y) and p 2 (χ, y) ar indpndnt. Also assum that all th funds ar global and thr is no spcializd fund. Lt N b th mass of informd managrs. Suppos p (χ, y) = (p, p 2 ) (p, ) 2 for th following valus of (χ, y); (χ, χ 2 ) = (0, 0) and (y, y 2 ) [y, y N ] 2 (χ, χ 2 ) = (0, ) and (y, y 2 ) [y, y N ] [y + N, y]. (χ, χ 2 ) = (, 0) and (y, y 2 ) [y + N, y] [y, y N ]. (χ, χ 2 ) = (, ) and (y, y 2 ) [y + N, y] 2 Suppos, (χ, χ 2 ) = (0, ). Th probability of p (χ, y) = (p, p 2 ) is quivalnt to th probability that (y, y 2 ) [y, y N ] [y + N, y] and is qual to N ( y y ) 2. But not that P r p (χ, y) = p, p 2 (χ, y) = p 2 χ, χ 2 is qual N y y j to ( ) 2 for any (χ, χ 2 ). Furthrmor, P r p i (χ, y) = p i χ i, χ j, p = p j = N for any valu of χ i, χ j and p j (p, y y ). Hnc, by Dfinition 5, p and p 2 ar indpndnt of ach othr. 5
Whn prics ar indpndnt, conditional on χ th probability of p b ing unrvaling is indpndnt of p 2 and χ 2. Thrfor, if p is unrvaling, uninformd managrs at all funds know that pric of asst 2 rvals nothing about th stat of asst. Obsrving p (χ, y) = p and p 2 (χ, y) = p 2, uninformd managrs at all funds try to figur out th probability of th rpaymnt of th assts by larning th actions of informd managrs of both spcializd and global funds. As long as th mass of 3 funds is non-zro, uninformd managrs form thir postriors about asst i taking into account both p i and p j. call that f j (p, p 2 ) P r(χ j = 0 p < p <, p < p 2 < ), i.., f j (p, p 2 ) is th postrior of th uninformd managrs funds aftr obsrving a pric pair (p, p 2 ). Nxt rsult shows that prics ar intrdpndnt in any quilibrium. Proposition 3. As long as thr ar som global funds in th markt, prics ar intrdpndnt in any quilibrium. To undrstand th intuition bhind this rsult, suppos asst 2 dfaults and p 2 is unrvaling. Whn asst rpays, informd managrs of global funds and informd managrs of funds spcializing in markt ar all dmanding asst. Howvr, informd managrs of global funds can dmand ithr asst or asst 2 whn both assts rpay. So whn asst rpays and asst 2 dfaults th dmand for asst is highr than whn both assts rpay. This implis that th probability that p = and th rpay of asst is rvald is highr whn asst 2 dfaults and p 2 is unrvaling. Thrfor, dfault or rpay of asst 2 changs th probability that p is rvaling th rpay of asst. Thrfor, p can not b indpndnt of th rpay or dfault of asst 2 and p 2. But sinc prics ar intrdpndnt by th dmands of N managrs, 3 uninformd managrs fac an advrs slction problm. Whn prics ar unrvaling, uninformd managrs rciv asst with a lowr probability whn asst 2 dfaults and with a highr probability whn asst 2 rpays or asst is dfaulting. n quilibrium, pric of asst must compnsat uninformd managrs for this advrs slction problm and must dcras following any shock to th x-ant dfault probabilitis of both asst and asst 2. Whn q 2 incrass, th dfault of asst 2 is mor likly and pric of asst 2 suffrs. Also, th advrs slction problm in markt is mor svr bcaus th probability of rciving th rpaying asst dcrass. Hnc, pric of asst must also dcras to compnsat uninformd managrs for th risk of not rciving th rpaying asst. Whn thr is no global fund, thr is no advrs slction problm and thr is no co-movmnt. 6
As long as thr ar som global funds and th mass of N3 managrs is not zro, thir trads contain information rgarding both assts and in quilibrium markt claring prics rval this information to all uninformd managrs. Whn thr is no 3 fund- and no N3 managr- th pric of asst only contains th information rvald by th dmands of N managrs. Sinc N managrs nvr dmand asst 2, p has no information rgarding th rpay or dfault of asst 2. Whn th invstmnt stratgy is spcialization in on markt managrs ar valuatd only basd on th rturns of that particular markt. But whn th invstmnt stratgy is to sk invstmnt opportunity in as many markts as possibl managrs rturns ar compard with th highst rturn among all th markts. Thrfor, vn a small mass of global funds is nough to induc th rst of managrs in ( 2 ) funds to xtract information about markt from th actions of N3 managrs at markt 2. Notic that th intrdpndnc is amplifid by th continuation payoff of bing mployd. To s this bttr, lt p (χ, y) = (P, P 22 ), that is N and N 2 managrs ar marginal tradrs in quilibrium. This mans that and γ( f j ) P jj = (7) γ + (2f j )δωβw j W f j j = (2f j ) (8) γ P jj This prmium is similar to th rputational prmium in Gurriri and Kondor (202) and disappars as soon as W j = 0. Howvr, vnif W j = 0, p (χ, y) and p 2 (χ, y) ar still intrdpndnt. This is bcaus at (P, P 22 ), uniformd tradrs fac th sam signal xtraction problm of uniformd managrs. Hnc, th postriors of uninformd tradrs, f j, ar not indpn dnt of p i = P ii and th rpay or dfault of asst i. Thus, P jj ar functions of q and q 2 and any chang in q and q 2 shifts both P and P 22. Th following Corollary summarizs th discussion. Corollary. As long as thr ar som global funds in th markt, prics ar co-moving in any quilibrium following any shock to priors. 3.3 Simpl Equilibria n this sction w charactriz som simpl partially rvaling quilibria. Th following Lmma givs th postriors of uninformd managrs at any 7
simpl quilibrium with unqual unrvaling prics. n all th following N rsults assum r j = y j y. r 3 q 2 q r ( q )q 2 Lmma. Assum <. n any simpl quilibrium at which with positiv probability (p, p 2 ) = (p, p 2 ) whr p < p 2 < p <, postriors of uninformd managrs about risky assts at (p, p 2 ) ar givn as ( q )( r q 2 r 3 ) f = r ( q )q 2 r 3 (9) ( q 2 )( r ) f 2 = r ( q )q 2 r 3 (20) Nxt, w driv th postriors of uninformd managrs whn unrvaling prics ar qual. Whn unrvaling prics ar qual, informd managrs of 3 funds ar indiffrnt btwn risky assts whn both rpay. So w can assum that α fraction of thm only asks asst and α fraction of thm asks asst 2 whr α is dtrmind in quilibrium so that th postriors of uninformd managrs at p = (p, p) ar qual. Th following Lmma is giving ths qual postriors in an quilibrium with qual prics. r 3 q 2 q r ( q )q 2 Lmma 2. Assum >. n any simpl quilibrium at which with positiv probability (p, p 2 ) = (p, p); p < p < is ralizd th postrior blifs of uninformd managrs about risky assts ar qual and givn as follows ( q )( r 2 ( α )r 3 )( r r 3 (q 2 + ( q 2 )α )) f = ( q )G 0 (r, r 2, r 3, α ) + q G (r, r 2, r 3, α ) (2) ( q 2 )( r α r 3 )( r 2 r 3 + α ( q )r 3 ) f 2 = ( q )G 0 (r, r 2, r 3, α ) + q G (r, r 2, r 3, α ) (22) whr G 0 (r, r 2, r 3, α ) =( r 2 ( α )r 3 )( r r 3 (q 2 + ( q 2 )α )) (23) G (r, r 2, r 3, α ) =( r α r 3 )( r 2 r 3 ( q 2 + ( α )q 2 )) (24) and α is th solution to ( q )( r 2 ( α)r 3 )( r r 3 ( q 2 )α r 3 q 2 ) ( q 2 )( r αr 3 )( r 2 r 3 + α( q )r 3 ) = 0 (25) 8
Lmmas and 2 show clarly that at any non-rvaling pric, th postriors of uninformd managrs ar not th sam as thir priors. Th diffrnc btwn f j and q j is th information lakd to th markt at any quilibr rium. Not that Lmmas and 2 put mutually xclusiv conditions on 3 r so th postriors of uninformd managrs ar always wll dfind for any valu of N and N 3. n Sami and Brusco (204), w provd that as long as thr ar no spcializd funds in th markt, thr is no quilibrium with unqual unrvaling prics. Nxt proposition shows th xistnc of such quilibrium at this modl. n this quilibrium, th information that is rvald to th markt is not nough to convinc uniformd managrs that th probability of th rpay of asst 2 is as high as asst. r 3 q 2 q Proposition 4. Assum < and r ( q )q 2 (i). 3 < C M y (ii). min{, 2 } > C + M whr C and Car givn in th Appndix. Thr xists an quilibrium at which p (χ, y) taks th following valus; som rvlation; γ( f j ) p j = P jj = (26) γ + (2f j )δωβw j whr f and f 2 ar givn in Lmma. partial rvlation at which ithr pj p j or p j =. γ( q i ) p i = (27) γ + (2q i )δωβw full rvlation at which for ach i =, 2 w hav ithr p i p i or p i =. Morovr, p < P 22 < P < and marginal tradrs at (P, P 22 ) ar uninformd managrs of spcializd funds. Whn thr ar no spcializd funds, th dmands for any asst is only coming from th managrs of global funds. At p > p 2, uninformd managrs ar marginal tradrs; and again fac th sam advrs slction problm; th i 9
probability of rciving asst whn it dfaults or asst 2 rpays is highr than th probability of rciving it whn it rpays and asst 2 dfaults. Thrfor, an quilibrium with unqual non-rvaling prics xists only if w hav nough spcializd funds in th markt so that th dmands of N j managrs can clar th markt and mak thm marginal tradrs. Whn th informd managrs hird at 3 funds ar rlativly lss than r 3 q 2 q th informd managrs hird at funds (so that < ), th r ( q )q 2 invstmnts of N3 managrs ar not nough to chang th prior blif of uninformd managrs about asst bing th highst rpaying asst. Morovr, assumptions (i) and (ii) imply that 3 < min{, 2 }. This mans that th total invstmnts mad by global funds is lss than th invstmnts of spcializd funds. Whn th total invstmnts of global funds ar low, markts ar clard only if thr is positiv dmands from spcializd funds and quilibrium prics ar st by uninformd managrs of spcializd funds. Thus, prics do not contain as much information as thy would if thr wr mor informd managrs hird at 3 funds and th siz of 3 funds wr. Whn marginal tradrs ar uninformd managrs of spcializd funds and p P j = jj, maximum prics that N 3 managrs pay ar solvd from quations ()-(2). Not that N3 managrs continuation payoff of bing mployd, W 3, is lss than th continuation payoff of uninformd managrs of spcializd funds. This is bcaus if both risky assts rpay, N managrs 3 who buy th mor xpnsiv asst ar fird. Nvrthlss, N j,j =, 2, managrs buying th rpaying risky asst ar always rtaind. Thrfor, whn N j managrs ar marginal tradrs at p j thir payoff of buying risk lss bond is γ + δωf j βw j whil th payoff to N 3 managrs of buying risk fr bond is γ + δωf f 2 W 3. Clarly, at p j = P jj th payoff to N3 managrs of buying risklss bond is lss than th payoff of buying asst j, thrfor P j 3 > P jj and it is not optimal for N 3 managrs to b indiffrnt btwn risky assts and risk lss bond at p j = P jj. Th nxt rsult charactrizs th quilibrium whn N 3 is larg. Whn th siz of global funds is larg rlativ to spcializd funds, th dmands of N3 managrs clar th markt and th only possibl quilibrium is th on with qual unrvaling prics. n this quilibrium marginal tradrs ar uninformd managrs of global funds. Whn p = p 2 both assts ar paying th sam xpctd rturn, so managrs hird at 3 funds ar fird if thy buy th dfaulting risky asst or risk fr bond whn at last on of th assts 20
is rpaying. Thus, P j 3 now solvs γ ( f j )( P + δωβw ) = γ + δωf f 2 W (28) j3 3 3 whr f and f 2 ar givn in Lmma 2. Th following proposition summarizs this discussion. r 3 q 2 q Proposition 5. Assum that > and r ( q )q 2 b +b 2 (i). 3 > + M. max{b,b 2 } (ii). min{, 2 } M >. Thn, thr xists an quilibrium at which p (χ, y) taks th following valus; som rvlation; γ( f j ) p j = P j 3 = (29) γ + (f f 2 f j )δωβw whr P 3 = P 23 (p, ), and f and f 2 ar givn in Lmma 2. 3 j j partial rvlation at which ithr p p j or p = and, γ( q i ) p i = (30) γ + (2q i )δωβw full rvlation at which for ach i =, 2 w hav ithr p i p i or p i =. Morovr, marginal tradrs at P j 3 ar uninformd managrs of global funds. 3.4 Non-Simpl Equilibrium p to now w just focusd on simpl quilibria whr p only gts a uniqu valu in (p, ) 2. Suppos (χ, χ 2 ) = (0, 0). Suppos also that and 3 funds hav hird vry fw informd managrs. Thn, th mass of informd managrs in 3 funds is not nough to rval nough information to convinc th uninformd managrs to bid th sam pric for both assts in quilibrium. Morovr, for som valus of liquidity trading quilibrium prics do not rval any information to uninformd managrs and th postriors of 2 i
uninformd managrs ar th sam as thir priors, i.. f i = q i. As w discussd in Sami and Brusco (204) bcaus of advrs slction problm that ariss for marginal N3 managrs whn prics ar diffrnt, th quilibrium only xists if th marginal tradrs ar uninformd managrs of spcializd funds. Proposition 6. Suppos Assum also, (r + r 2 + 3r 3 ) + (r 2 + r 3 )[r 4(r + r 3 )] ( q )q 2 < (r + r 2 + 2r 3 ) 3(r + r 3 )(r 2 + r 3 ) q ( q 2 ) (i). 3 < C M y. (ii). min{, 2 } > C + M. Thr xists a partially rvaling, non-simpl quilibrium at which p (χ, y) taks th following valus; no rvlation; whr j =, 2. som rvlation; γ( q j ) p = P j jj (q j ) = (3) γ + (2q j )δωβw j γ( f j ) p j = P jj (f j ) = (32) γ + (2fj )δωβw j whr j =, 2, and f j is givn in th appndix. partial rvlation at which ithr p or p = and, j p j j γ( q i ) p i = (33) γ + (2q i )δωβw full rvlation at which for ach i =, 2 w hav ithr p i p i or p i =. Morovr, marginal tradrs at P ar uninformd managrs of spcializd jj funds. i 22
p ((χ, y)) and p 2 ((χ, y)) co-mov bcaus at (P (f ), P 22 (f 2)), th probability of p = P j jj is not indpndnt of χ i and pi = P ii. Howvr, whn no information is rvald through prics at (P (q ), P 22 (q 2)), th probability that p = P (q ) is indpndnt of th rpay or th dfault of asst 2 and p = P 22. Pric co-movmnts only disappar whn thr is no 3 fund in th markt. But in that cas, lss information laks to th markt as wll. 3.5 Optimal tntion ul Th optimal bhavior of funds in our modl is idntical to th on in Gurriri and Kondor (202). W hav to prov that th firing rul of funds ar optimal. Spcializd funds fir thir managrs whn thy buy th dfaulting asst or risklss bond whn asst rpays. Global funds fir th managrs if thy don t achiv th highst x-post rturn. Givn that th rturn signals to informd managrs ar prfct, any managr with wrong invstmnt dcision is immdiatly rvald uninformd with probability. f th prcntag of th informd managrs in unmploymnt pool is always non-zro, it is optimal to fir an uninformd managr. But rcall that a fraction δ of informd managrs is always sparatd from th funds. Sparatd or unmployd informd managrs always sarch for a job bcaus by fr ntry condition for uninformd managrs, informd managrs gt a positiv xpctd pay-off if thy look for a job. Thus, unmploymnt pool is nvr mpty of informd managrs and it is optimal to fir a managr that is rvald uninformd. t rmains to show that funds rtain a managr who has mad th right invstmnt dcision and is not rvald uninformd by xognous signal. This is th cas whn th updatd blif of funds about managr bing informd is highr than th probability that a just hird managr is informd, i.., L t η t+ > t = (34) L + L But (34) holds givn th assumption ω >, by xactly th sam argu +δ mnts in th proof of Proposition of Gurriri and Kondor (202) and Proposition 7 of Sami and Brusco (204) and assuming. This assumption nsurs that whn a managr is not rvald uninformd and has not mad any mistak th blifs of funds improvs with a high nough spd that surpasss th probability of hiring an informd managr from th unmploymnt pool. t t 23
4 Conclusion This papr discussd pric co-movmnt btwn two financial markts in a risk nutral world with indpndnt liquidity and rturn shocks. Th invstmnt dcisions of funds ar dlgatd to fund managrs who ar informd or uninformd on th rturn of th assts and fac dismissal if thy don t mak th highst possibl rturn. W showd that in any quilibrium of th modl prics co-mov with ach othr following a shock to th priors on any asst. n quilibrium, markt claring prics rflct all th information availabl in th markt. As long as thr ar som global funds in th markt, th dmands of informd managrs hird at ths funds rval information about both assts. Whn th total mass of informd managrs is so low that markt is not clard, th quilibrium pric must mak uninformd managrs marginal tradrs. But as long as thr ar som global funds in th markt, with highr probability uninformd managrs rciv th dfaulting asst or riskfr bond whn both prics ar unrvaling. n quilibrium, prics must compnsat uninformd managrs for this advrs slction problm and ar functions of th x-ant dfault probabilitis of both assts. Hnc, any shock to th x-ant dfault probability of on asst changs th pric of both assts. Th rputationally concrnd managrs always ask for a prmium ovr th risk fr rat that compnsats thm for th risk of bing dismissd. This prmium magnifis th co-movmnt btwn prics. Howvr, th comovmnt dosn t disappar if thr is no rpuatational concrn. This mans that vn if invstors wr dirctly invsting thir capital and wrn t dlgating th invstmnt dcision to fund managrs, prics would still co-mov with ach othr. Co-movmnt only disappars whn thr is no global fund in th markt. But if thr is no global fund, lss information is rvald to th markt. This suggsts that thr is a trad-off btwn markt stability and information rvlation. Global funds incras th pric co-movmnt but rval mor information to th markts. Without global funds thr is no pric co-movmnt and mor markt stability, but lss information rvlation as wll. 24
5 Appndix n all th following rsults assum q 2 > q. Not that this lads to P 22 < P. Bsids, not that by (9), in th quilibrium with unqual prics f 2 > q 2 and f < q, hnc, γ( q ) P < P = (35) γ + δωβ(2q )W γ( q 2 ) ( q 2 )( M Δy ) 2 P 22 > > P = (36) γ + δωβ(2q2 )W γ + δωβw 2 3 max{b,b 2 } Lt C = tion as follows; and C = min{b, b 2 }P. W rwrit th supply assump- M + y < min{b, b 2 }.P (37) Claim. Suppos p (χ, y) is an quilibrium pric mapping. Dfin { } = (p, p 2 ) (p, ) 2 p > p 2, p (χ, y) = (p, p 2 ) for som (χ, y) Also, dfin Φ (χ,χ 2 ) = {y = (y, y 2 ) p (χ, y) = (p, p 2 ) for som (p, p 2 ) } (38) Thn, (A). Φ 00 = {(y, y 2 ) y y y N, y y 2 y N 2 N 3 }. (A2). Φ 0 = {(y, y 2 ) y y y N N 3, y + N 2 + N3 y 2 y}. (A3). Φ 0 = {(y, y 2 ) y + N y y, y y 2 y N 2 N 3 }. (A4). Φ = {(y, y 2 ) y + N y y, y + N 2 + N3 y 2 y} Proof. (i). Lt y [y, y N ] [y, y N 2 N 3 ], thn z i ((0, 0), (y, y 2 )) = z i ((, ), (y +N, y 2 +N 2 +N 3 )). Thus, by blif consistncy condition w must hav p i ((0, 0), (y, y 2 )) = p i ((, ), (y + N, y 2 + N 2 + N 3 )). But this is possibl only if quilibrium prics ar unrvaling and p 2 p. Thrfor, [y, y N ] [y, y N 2 N 3 ] Φ 00. Lt (y, y 2 ) Φ 00 but (y, y 2 ) / [y, y N ] [y, y N 2 N 3 ]. This mans that z 2 ((0, 0), (y, y 2 )) y which rvals rpay for asst 2 and hnc p 2 must b /. Contradiction. 25
(ii). Lt y [y, y N N 3 ] [y + N + N 3, y], thn z i ((0, ), (y, y 2 )) = z i ((, ), (y + N + N 3, y 2 )). Thus, by blif consistncy condition w must hav p ((0, ), (y, y 2 )) = p ((, ), (y + N + N i i 3, y 2 )) which is possibl only if prics ar unrvaling. Thrfor, [y, y N N 3 ] [y + N + N 3, y] Φ 0. f (y, y 2 ) Φ 0 /[y, y N N 3 ] [y + N + N 3, y], thn y > y N N3 and z ((0, ), (y, y 2 )) > y which implis p = /. Contradiction. (iii). Similar to (2). (iv). Lt y [y + N, y] [y + N + N 2 3, y], thn z i ((, ), (y, y 2 )) = z i ((0, 0), (y +N, y 2 +N +N 3 )) and by blif consistncy p i (, ), (y, y 2 )) = p i ((0, 0), (y + N, y 2 + N + N 3 )) which is possibl only if prics ar not rvaling. So [y + N, y] [y + N 2 + N 3, y] Φ. f y Φ /[y + N, y] [y + N + N 2 3, y], thn y < y + N and z ((, 0), (y, y 2 )) < y + N which implis that N managrs havn t dmandd asst. This only happns whn asst is dfaulting so p = p which is a contradiction. Claim 2. Suppos p (χ, y) is an quilibrium pric mapping. Dfin Also, dfin Thn, { } = (p, p) (p, ) 2 p (χ, y) = (p, p), for som (χ, y) Φ ' = {y = (y, y 2 ) p (χ, y) = (p, p) for som (p, p) } (39) (χ,χ 2 ) (i). Φ ' = {y = (y, y 2 ) y y y (N + αn 3 ), y y 2 y N 00 2 ( α)n 3 }. (ii). Φ ' = {(y, y 2 ) y y y N N 3, y + N 0 2 + ( α)n3 y 2 y}. (iii). Φ ' = {(y, y 2 ) y + N + αn y y, y y 2 y N 2 N 0 3 3 }. (iv). Φ ' = {(y, y 2 ) y +N +αn y y, y +N 3 2 +( α)n3 y 2 y}. whr α is th fraction of N 3 that buy asst whn p (χ, y) = (p, p). 26
Proof. Similar to th proof of Claim. Lmma. Not that by Claim P r((p, p 2 ) (p, )) = ( q )( q 2 )P r((y, y 2 ) Φ 00 ) + ( q )q 2 P r((y, y 2 ) Φ 0 ) +q ( q 2 )P r((y, y 2 ) Φ 0 ) + q q 2 P r((y, y 2 ) Φ ) (40) Thrfor, ( q )( r q 2 r 3 ) f = r ( q )q 2 r 3 (4) ( q 2 )( r ) f 2 = r ( q )q 2 r 3 (42) r 3 q r t is clar that 0 < f <. Sinc, > r q 2 q Also sinc 3 <, f > f 2. r ( q )q 2, 0 < f 2 < as wll. Proof of Lmma 2. call that if in quilibrium p < p i < /, marginal tradrs ar ithr uninformd managrs hird at 3 funds or uninformd managrs at j funds. Thn p = p 2 = p is only possibl if postriors of uninformd managrs ar also qual. f informd managrs of global funds ar indiffrnt btwn th risky assts whn both rpay and hav th sam pric, th only strict dmand for any asst is coming from th informd managrs in spcializd funds. But in this cas, th postriors ar not qual. Thus, w nd to hav α fraction of N managrs dmanding asst and 3 ( α ) of thm dmanding only asst 2 whn both assts rpay and hav th sam pric whr α is dtrmind in quilibrium to quat th postriors on risky assts. Nxt, not that 2 2 P r(p = p, p = p, χ i = 0) f i = P r(χ i = 0 p = p, p = p) = (43) P r(p = p, p 2 = p) sing Claim 2, 2 2 P r(p = p, p = p, χ = 0) = ( q )( q 2 )P r((y, y 2 ) Φ 00 ) + ( q )q 2 P r((y, y 2 ) Φ 0 ) P r(p = p, p = p) = ( q )( q 2 )P r((y, y 2 ) Φ 00 ) + ( q )q 2 P r((y, y 2 ) Φ 0 ) + q ( q 2 )P r((y, y 2 ) Φ 0 ) + q q 2 P r((y, y 2 ) Φ ) (44) 27
Thrfor, ( q )( r 2 ( α )r 3 )( r r 3 ( q 2 )α r 3 q 2 ) f = ( q )G 0 (r, r 2, r 3, α ) + q G (r, r 2, r 3, α ) (45) ( q 2 )( r α r 3 )( r 2 r 3 + α ( q )r 3 ) f 2 = ( q )G 0 (r, r 2, r 3, α ) + q G (r, r 2, r 3, α ) (46) whr G 0 (r, r 2, r 3, α ) =( r 2 ( α )r 3 )( r r 3 + ( q 2 )( α )r 3 ) (47) G (r, r 2, r 3, α ) =( r α r 3 )( r 2 r 3 + q 2 α r 3 ) (48) Sinc w must hav qual postriors, α is th solution to H(α) ( q )( r 2 ( α)r 3 )( r r 3 ( q 2 )α r 3 q 2 ) ( q 2 )( r αr 3 )( r 2 r 3 + α( q )r 3 ) = 0 (49) Not that, H(α = ) > 0 by q 2 > q and H(α = 0) < 0 by assumption. Thus, thr xists 0 < α < at which f = f 2. Proof of Proposition 3. At any simpl quilibrium with unqual prics, P r(p = p, p2 = p 2 (χ, χ 2 ) = (0, 0)) = P r(φ 00 ) = ( r )( r 2 r 3 ) (50). But, P r(p = p χ = 0) = ( r )( q 2 ) + ( r r 3 )q 2 (5) P r(p 2 = p 2 χ 2 = 0) = ( r 2 r 3 )( q ) + ( r 2 r 3 )q = ( r 2 r 3 ) (52) Th probabilitis in (50)-(52) show that at any simpl quilibrium with diffrnt unrvaling prics, p and p 2 ar intrdpndnt. Th sam argumnt togthr with th us of Claim 2 shows that at any simpl quilibrium with qual prics, p and p 2 intrdpndnt. Suppos p (χ, y) is a non-simpl quilibrium pric vctor. Dfin P = n i i { i χχ i= as a partition of φ χ χ 2, p 2 ) (p, ) 2 2 } and suppos p (χ, y) = (p ; (χ, y) and p pi i i p 2, for any (y, y 2 ) χ χ 2. Also assum that p 2 (χ, y) ar indpndnt. Thrfor, for any i =,..., n, P r(p (χ, y) = (p i, p i i i 2 ) χ = 0, χ 2 = 0) = P r(p (χ, y) = p χ = 0)P r(p 2 (χ, y) = p 2 χ 2 = 0) (53) 28