Opiml Conrcs in Coninuous-ime Deleged Porfolio Mngemen Problem Hui Ou-Yng Duke Universiy nd Universiy of Norh Crolin his ricle sudies he conrcing problem beween n individul invesor nd professionl porfolio mnger in coninuous-ime principl-gen frmework. Opiml conrcs re obined in closed form. hese conrcs re of symmeric form nd sugges h porfolio mnger should receive fixed fee, frcion of he ol sses under mngemen, plus bonus or penly depending upon he porfolio s excess reurn relive o benchmrk porfolio. he pproprie benchmrk porfolio is n cive index h conins risky sses where he number of shres invesed in ech sse cn vry over ime, rher hn pssive index in which he number of shres invesed in ech sse remins consn over ime. Fund mngers compension schemes in deleged porfolio mngemen seing hve been of specil ineres o boh cdemics nd prciioners. Acdemic ineres sems in pr from he rpid growh of mnged funds over he ls wo decdes. Gompers nd Merick (1998) repor h by December 1996, muul funds, pension funds, nd oher finncil inermediries held discreionl conrol over more hn hlf of he U.S. equiy mrke. I is hus of impornce o sudy he impc of insiuionl rding on sse prices nd o inegre ino one model boh sse pricing nd deleged porfolio mngemen compension, s dvoced by Brennn (1993), Allen nd Snomero (1997), nd Allen (21). I is lso imporn o undersnd he relionship beween mnger s rding sregies nd his implici incenives or creer concerns see, e.g., Brown, Hrlow, nd Srks (1996), Brown, Goezmnn, nd Prk (1997), Chevlier nd Ellison (1999), nd Chen nd Penncchi (2)]. hese concerns include inernl promoion from smll o lrge fund, higher ouside offers, nd he flow of money hrough he fund under mngemen. 1 he impornce of fund mnger compension is lso underscored by Congressionl pssge in 197 of he Amendmen o I m very greful o Hu He for offering mny insighful suggesions nd for invluble guidnce. I m lso very greful o boh Domenico Cuoco nd he referee for poining ou n error nd for offering mny insighful commens nd suggesions h hve improved he ricle immensely. Mny hnks go o M Spiegel, Henry Co, Peer DeMrzo, Rob Hnsen, Hyne Lelnd, nd Mrk Rubinsein for vluble commens nd dvice, nd o Gry Goron for his encourgemen nd pience. I lso hnk Dong-Hyun Ahn, Nvnee Aror, ony Ciochei, Snjiv Ds, Bin Go, Diego Grci, Nengjiu Ju, Pee Kyle, Krl Lins, Jim McCrhy, om Noe, Seve Slezk, nd Nel Soughon for heir dvice. his ricle is revised version of Chper 2 of my Ph.D. disserion submied o he Hs School of Business, Universiy of Cliforni Berkeley. Address correspondence o Hui Ou-Yng, Fuqu School of Business, Duke Universiy, Durhm, NC 2778-12, or e-mil: huiou@duke.edu. 1 For exmple, Chevlier nd Ellison (1999) find h due o muul fund mngers creer concerns, younger mngers ke on lower unsysemic risk nd devie less from ypicl behvior hn heir older counerprs. he Review of Finncil Sudies Spring 23 Vol. 16, No. 1, pp. 173 28 23 he Sociey for Finncil Sudies
he Review of Finncil Sudies /v 16 n 1 23 he Invesmen Advisors Ac of 194 (herefer he Amendmen ), which limis he ypes of compension schemes h my be used by muul funds, pension funds, nd oher publicly regisered invesmen compnies. Of priculr ineres is he requiremen h when performnce-bsed incenive fees involving benchmrk re used, hen hey mus be symmeric round he chosen benchmrk. hus if n invesmen compny is llowed o receive bonus when is porfolio reurn is bove he reurn on he benchmrk porfolio, hen i mus lso receive penly when is porfolio reurn is below he reurn on he benchmrk porfolio. 2 Alhough performnce-bsed compension schemes hve become quie populr in recen yers, here is considerble debe mong cdemics over wheher he symmeric incenive fee now required by lw is in fc economiclly jusified bsed on modern finncil economics. 3 In his ricle we derive opiml porfolio mngemen conrcs in closed form in relively simple economy. Previous nlyses of he principl-gen problem hve no emped o produce n opiml conrc from conrc spce h covers he symmeric incenive fees. Moreover, due o he bsence of he sock prices in prior models, benchmrk porfolios, such s pssive index, in which he number of shres invesed in ech sse remins fixed over ime, re no cpured in heir conrc spce. As resul, i hs no been possible, using prior models, o exmine he fundmenl quesion of wheher he symmeric compension scheme prescribed by he Amendmen for fund mngers is efficien in generl principl-gen frmework. Furhermore, i hs no been possible o more horoughly ddress issues regrding he impc of deleged porfolio mngemen on equilibrium sse prices, nor he impc of fund mngers creer concerns on heir rding sregies. We hus emp o provide n economic foundion h my be used for he ssessmen of he symmeric incenive fees, like hose required by he Amendmen, for he sudy of he impc of mngers creer concerns on heir rding behvior nd for he developmen of sse pricing models in he presence of deleged porfolio mngemen. o do so, we nlyze he relionship beween n individul invesor nd professionl porfolio mnger in coninuous-ime principl-gen frmework. he invesor (principl) enruss her funds o nd provides conrc for he mnger (gen). In our opiml conrc soluion, boh he principl s nd he gen s dynmic mximizion problems re solved simulneously: he gen s porfolio choice 2 Prior o 197, symmeric incenive fees were lso used. Under his fee srucure, he mngemen would receive bonus if is porfolio ouperformed he benchmrk porfolio, nd no penly even if is porfolio underperformed he benchmrk. Noice h he Amendmen does no require funds o use ny benchmrks. I only requires h if fund uses n incenive fee involving benchmrk, he fee mus be symmeric round he benchmrk. 3 For deiled discussion on muul fund fee srucure, see, for exmple, Srks (1987), Grinbl nd imn (1989), Golec (1992), Brown nd Goezmn (1995), Ds nd Sundrm (1998), nd Cuoco nd Kniel (2). For compension schemes for hedge fund mngers, see, for exmple, Fung nd Hsieh (1997), Goezmnn, Ingersoll, nd Ross (1997), nd Ackermnn, McEnlly, nd Rvenscrf (2). 174
Opiml Conrcs in Porfolio Mngemen Problem depends upon he conrc ssigned by he principl, nd he principl s conrc kes ino ccoun he gen s porfolio choice. Our opiml conrcs re obined from lrge conrc spce nd hey re shown o be of symmeric form. hey sugges h fund mnger should be pid fixed fee, frcion of he ol sses under mngemen, plus bonus or penly depending upon he excess reurn on he porfolio. We lso show h he pproprie benchmrk is n cive index in which he number of shres invesed in ech sse chnges over ime, rher hn he pssive index. For rcbiliy of he model, we ssume h he mnger s uiliy funcion is of negive exponenil form. When he invesor lso hs n exponenil uiliy funcion, opiml conrcs re obined in closed form using clss of cos funcions. If insed generl uiliy funcion for he invesor is doped, hen he invesor s mximizion problem becomes nlyiclly inrcble in he presence of generl cos funcion. o provide insigh ino our model in h cse, we re specil cse in which he mnger s cos funcion is consn. Opiml conrcs re hen derived in closed form. Our model cn be inerpreed s sndrd principl-gen model in which he gen s cion ffecs boh he drif (i.e., he reurn) nd he diffusion res (i.e., he risk) simulneously, hus represening n exension of he previous principl-gen lierure. Our nlyses of he porfolio mngemen problem show h he mnger s cion or his porfolio policy ppers in boh he drif nd he diffusion erms. Previous coninuous-ime principl-gen models such s he well-known Holmsröm nd Milgrom (1987), Schäler nd Sung (1993), nd Sung (1995) models cnno be pplied o he deleged porfolio mngemen problem becuse hey do no llow he gen s cion o influence boh he drif nd he diffusion erms simulneously. 4 In Holmsröm nd Milgrom nd Schäler nd Sung, for exmple, he gen conrols he drif re lone. And in Sung, he gen conrols he drif nd diffusion res seprely, nd here re wo independen conrol vribles, so h he diffusion re is held consn when he gen conrols he drif re, nd vice vers. he res of he ricle is orgnized s follows. We briefly review he reled heoreicl lierure on he deleged porfolio problem in he nex secion. Secion 2 describes he bsic seup. Secion 3 derives n expression for he opiml compension in erms of he gen s vlue funcion. In Secions 4 nd 5, wo ypes of problems re reed nd opiml conrcs re obined. Some concluding remrks regrding he model re offered in Secion 6. Appendix A shows h under regulriy condiion, he Bellmn equion is boh necessry nd sufficien condiion for dynmic mximizion problem. Proofs of Proposiion 1, of heorem 1 nd Corollry 1, nd of Proposiion 2 nd heorem 2 re presened in ppendixes B, C, nd D, respecively. 4 Bolon nd Hrris (1997) nd Deemple, Govindrj, nd Loewensein (21) exend he Holmsröm Milgrom model o include more generl oupu processes. 175
he Review of Finncil Sudies /v 16 n 1 23 1. Reled Lierure In his secion we briefly review he heoreicl reserch on he deleged porfolio mngemen problem, which flls ino hree cegories nd ypiclly kes conrc forms s exogenously given. he firs cegory ries o ddress he implicions of n pproprie benchmrk wihin he symmeric conrc form prescribed by he Amendmen. Wihin liner nd symmeric conrc form, Admi nd Pfleiderer (1997) show in one-period seing h he use of benchmrk porfolio of risky sses such s n index fund cnno be esily rionlized. Since heir objecive is no o sudy opiml conrcs, no lernive benchmrks re offered. he Admi Pfleiderer resul chllenges he common prcice where porfolios of risky sses re ofen doped s benchmrks for incenive fees nd, s resul, i bers poenilly significn policy implicions. However, i is imporn o exmine he issue regrding pproprie benchmrks in muliperiod relionship s sudied in his ricle, becuse his enlrges he spce for benchmrk porfolio of risky sses over wh is possible in oneperiod model. he lrger spce includes no only pssive index bu lso n cive index. Noice h he cive index cnno be cpured in one-period model in which he mnger invess only once. 5 he second cegory of deleged porfolio mngemen reserch compres he symmeric nd symmeric incenive fee srucures. For exmple, Srks (1987) nlyzes nd compres he symmeric nd symmeric conrcs nd shows h he symmeric conrc domines he symmeric one in inducing he mnger o choose he porfolio policy desired by he invesor. In oneperiod nd hree-se risk-shring model, Ds nd Sundrm (1998), however, find lile jusificion for he symmeric conrc nd develop condiions under which he symmeric conrc provides Preo-dominn oucome. Soughon (1993) exmines he impc of he liner nd qudric conrcs on boh he invesor s welfre nd he mnger s effor o cquire informion. He shows in limiing cse h he qudric conrc my llow he invesor o chieve he firs-bes resul in morl hzrd model. he hird cegory of reserch exmines he impc of given symmeric nd symmeric fee srucures on he mnger s porfolio policy nd sse prices in n equilibrium seing. For exmple, on one hnd, Grinbl nd imn (1989) show h given n opion-ype symmeric conrc, if he mnger cn hedge his compension, hen he would choose invesmen sregies h increse he fund s voliliy. On he oher hnd, Crpener (2) shows h if he mnger cnno hedge, hen he opion compension my no necessrily led o greer risk seeking. Assuming h he mnger s compension 5 Previous nlyses of he gency relionship beween n invesor nd mnger hve ssumed discree-ime (ofen one-period) relionship. See, for exmple, Bhchry nd Pfleiderer (1985), Dybvig nd Sp (1986), Cohen nd Srks (1988), Kihlsrom (1988), Allen (199), Soughon (1993), Heinkel nd Soughon (1994), Dow nd Goron (1997), Lynch nd Muso (1997), Grci (2), Jing (2), Wonder (2), nd Core nd Qin (21). See lso Mmysky nd Spiegel (21) for differen pproch o he issue of muul funds. 176
Opiml Conrcs in Porfolio Mngemen Problem is bsed on performnce relive o n index, Brennn (1993) exmines he expeced sock reurns in equilibrium for one-period men-vrince economy nd finds h he choice of he benchmrk porfolio ffecs he equilibrium srucure of expeced reurns. Cuoco nd Kniel (2) nlyze he implicions of compension scheme h includes boh he symmeric nd he symmeric srucures. hey find h symmeric performnce fees induce significn posiive effec on he equilibrium prices of socks included in he benchmrk porfolio, significn negive effec on heir equilibrium Shrpe rios, nd mrginlly posiive effec on heir equilibrium voliliies. hey lso find h symmeric fees cn genere he opposie pricing implicions. 2. he Bsic Seup Consider simple economy in which principl hires n gen o mnge her porfolio. For exmple, he principl my represen ll he invesors of muul or pension fund, nd he gen hen represens he fund compny. 6 he ime horizon is ken o be. he principl invess W wih he gen =, nd he vlue of his invesmen is W =.For rcbiliy, we ssume h he principl cnno wihdrw funds from or dd funds o he porfolio. In he cse of pension fund, erly wihdrwls incur hevy penlies nd new conribuions ypiclly ke plce on cerin des, such s pydys. In he cse of muul fund, his ssumpion holds well for one-dy conrc period becuse he fund inflows nd ouflows occur only he end of he dy. For longer conrc period, his ssumpion implies h he fund inflows offse he fund ouflows during he conrc period or h here re no ne fund flows. here is one riskless bond nd N risky socks vilble for he gen o rde ny ime beween nd. Assume h he price B of he riskless bond follows deerminisic process db = rb d nd h he ineres re r is consn. Also ssume h he price process P i for ech risky sock h pys no dividends is described by he following geomeric Brownin moion: dp i = P i i d + P i i db where i is consn, i is he ih row of consn mrix in R N d wih linerly independen rows, nd he rnspose of B, B B 1 B d, d N, is sndrd Brownin moion defined on complee probbiliy spce (, P,F ). Ech specifies complee hisory of he Brownin moion. We shll wrie he N sock price processes in compc form s dp = dig P d + db, where P nd re wo N 1 vecors 6 Here we hve omied he poenil conflici beween he fund compny nd he mnger hired by he compny o cively mnge he fund. We shll use principl (gen) nd invesor (mnger), nd fund compny nd mnger inerchngebly. 177
he Review of Finncil Sudies /v 16 n 1 23 nd where dig P represens n N N mrix wih he digonl erms being P 1 P 2 P N, nd he off-digonl erms being zero. I is well known h he welh process W for he porfolio is given by see, e.g., Meron (1969, 1971), Ingersoll (1987), nd Duffie (1996)] dw = rw + A h d + A db (1) where h r1 nd where A A 1 A N denoes he dollr moun invesed in he risky socks nd is shor-hnd noion for funcion of, W s, nd P s, wih s. 7 he mnger decides how much o inves in bonds nd socks nd coninuously djuss his porfolio posiions. he conflic my rise becuse he invesor does no observe he mnger s porfolio policy vecor A. We here focus upon he incenive conflic beween n invesor nd mnger where here is no symmeric informion bou sock reurns. Jusificions for he employmen of porfolio mnger my include he mnger s lower rnscion coss on socks, he invesor s desire for diversified porfolio, nd he invesor s lck of ime for cive invesmen. hough he belief h mnger my possess superior informion is n imporn reson for his employmen, he finding h cively mnged muul funds, on verge, underperform pssive index funds my rise quesions on he vlidiy of his belief. 8 In ddiion, mnger s informion bou limied number of individul socks will no ply mjor role if n invesor is only ineresed in sse llocions mong money mrke fund, diversified domesic index porfolio, nd diversified index porfolios of foreign counries. If he invesor observes boh he sock price vecor P nd he welh process W of he porfolio coninuously, hen she cn infer precisely he mnger s porfolio policy vecor A from he fc h he insnneous covrince beween W nd P equls dig P A. Since is inverible by ssumpion, he mnger s policy vecor A is compleely deermined. 9 Hence we mus ssume h he invesor does no observe he welh nd he sock price processes simulneously. In prcice, invesors do no know muul or pension fund s minue-by-minue vlue nd cn only observe he vlue of he fund he end of he dy. On he oher hnd, i is relively esier o rck down sock prices. herefore we ssume h 7 echniclly, n dmissible A mus be progressively mesurble wih respec o F nd mus sisfy E A 2 d <,.s. 8 he mjoriy of sudies now find h cively mnged funds, on verge, provide lower ne reurns hn pssively mnged indexes. See, for exmple, Lkonishok, Shleifer, nd Vishny (1992), Elon e l. (1993), Gruber (1996), nd Crhr (1997). On he oher hnd, Grinbl nd imn (1989, 1993) nd Wermers (2) find h hough cively mnged muul funds underperform he mrke indexes on ne reurn bsis, hey my ouperform heir benchmrks before expenses re deduced. Asymmeric informion my ply more prominen role in he cse of hedge funds. Recen heoreicl work by Dybvig, Frnsworh, nd Crpener (2), Grci (2b), nd Sung (2) my provide insigh ino he undersnding of deleged porfolio problem in he presence of symmeric informion bou sock reurns. 9 We re greful o he referee for his poin. 178
Opiml Conrcs in Porfolio Mngemen Problem he invesor observes he sock price processes coninuously over ime nd h she observes he erminl vlue of he porfolio only. 1 Consequenly he conrc my depend upon he vlues of P, nd W. hroughou his ricle we ssume h he gen s preference over welh is described by U = 1 exp R R W. Here signifies he gen, nd R nd W denoe, respecively, he consn risk version coefficien nd he erminl welh for he gen. he principl s uiliy funcion is minly consrined o be negive exponenil, bu my be of generl form in specil cse. A ime, he principl offers he gen conrc S P W,. Assume h he conrc spce S P W is of he following form: S P W = E + W P d + 1 W P dw + 2 W P dp (2) where E is generl funcion of, W, nd P, nd where W, P, 1 W P, nd 2 W P re unknown coefficiens h my depend upon, W, nd P. 11 Noe h 2 nd P denoe row nd column vecor, respecively. his conrc spce covers no only ll wice coninuously differenible funcions of W nd P, bu lso ph-dependen funcions of P excep h, 1, nd 2 ime do no depend upon he ps hisory of P. he conrc spce lso includes ph-dependen funcions of W, bu we shll exclude from our soluion conrcs h re funcions of W. I is cler h his conrc form includes generl porfolio of risky sses s benchmrk. For exmple, if 2 represens negive consn vecor, hen he benchmrk porfolio is pssive index. S P W is pyble only ime. herefore he principl s erminl welh is W S P W nd he gen s erminl welh is S P W minus he coss ssocied wih mnging he porfolio. If S P W ffords he gen les his reservion uiliy, 1 R exp R E, ime, he underkes he job wih he undersnding h he my no qui i ny ime beween nd. Here E denoes he gen s ceriny equivlen welh ime. 1 Unlike he gen s effor in rdiionl principl-gen problem, he mnger s rding records cn be verified ex pos. Alernively we my simply ssume h forcing conrcs re no fesible or h he invesor s conrc mus induce he mnger o choose volunrily opiml rding sregies. 11 he firs hree erms re dped from Schäler nd Sung (1993). We hnk he referee for suggesing he ls erm, which cpures he symmeric incenive fee srucure llowed by he Amendmen. 179
he Review of Finncil Sudies /v 16 n 1 23 We re now in posiion o formlly se he gen s nd he principl s mximizion problems. Given he conrc S P W, he gen s problem is o mximize his own expeced uiliy over his erminl welh: sup E A 1 R exp { R S P W ]}] c A W d s dw = rw + A h d + A db (3) where c denoes he monery cos re incurred by he gen. he principl s objecive is o choose n S P W nd n A vecor so s o mximize her expeced uiliy over he erminl welh subjec o vrious consrins: sup E U p W S P W S P W A s s dw = rw + A h d + A db E 1 ]}] exp { R R S P W c A W d 1 R exp R E (4) nd subjec o he gen s incenive compibiliy consrin h he principl s opiml policy A mus lso solve he gen s mximizion problem. he inequliy represens he gen s pricipion consrin. I is binding he principl s opiml soluion; oherwise he principl could lwys lower S P W while sill geing he gen o ccep he job. he soluions o he mximizion problems in Equions (3) nd (4) shll be referred o s he second-bes soluions. Noice h he wo mximizion problems re no independen; he principl s conrc depends upon he gen s cion nd he gen s cion is bsed upon he principl s conrc. Consequenly one mus solve hese wo problems ogeher o chieve opiml resuls. 3. An Expression for he Opiml Fee Srucure In his secion we provide n expression for he opiml or equilibrium fee srucure o be denoed by S in erms of he gen s vlue funcion, 1, nd 2, using he gen s dynmic mximizion problem nd his pricipion consrin. he opiml fee S represens only he equilibrium moun h he principl pys o he gen if he gen dops he principl s opiml policy. S my or my no implemen he principl s opiml policy. he opiml conrc S P W, however, mus implemen he opiml policy. S P W nd S re equl in equilibrium. I shll be seen h S sisfies he gen s pricipion consrin nd grely simplifies he principl s mximizion problem. 18
Opiml Conrcs in Porfolio Mngemen Problem We begin wih he gen s mximizion problem. Define vlue funcion process V W P V W P for he gen s problem s 12 V W P = sup E 1 exp A u R + { R E + E 1 exp { R R E + + + 1 u W P dw u + u W P du c u W A du ]}] u W P du 1 u W P rw + A h du + 2 u W P dp u c u W A du+ 1 u W P A db u 2 u W P dig P du 2 u W P dig P db u ]}] (5) where A u denoes he soluion o he mximizion problem in V W P. We show in Appendix A h under regulriy condiion, sisfying he following Bellmn-ype equion is boh necessry nd sufficien condiion for A o be n opiml soluion: { = sup A V W P R W P + 1 W P rw +A h + 2 W P dig P c W A 1 2 R2 1 W P A } + 2 W P dig P 1 W P A + 2 W P dig P +V +V W rw +A h R 1 W P A + 2 W P dig P A + 1 2 V WWA A+V P dig P R 1 W P dig P A R dig P dig P 2 W P + 1 ] 2 r V PP dig P dig P +V WP dig P A (6) In Equion (6) he subscrips denoe he relevn pril derivives of V W P. 12 For noionl convenience, we shll wrie W nd P rher hn W nd P in,, c, V, ec. 181
he Review of Finncil Sudies /v 16 n 1 23 Using he gen s pricipion consrin, his Bellmn equion, Io s lemm, nd rnsformion, we cn rrive n expression for he equilibrium fee, which is presened in he nex proposiion. Proposiion 1. he equilibrium fee S is given by S = E + cd + 1 ( ) VW 2R V R 1 A ( ) V 2 + P V R 2 dig P d 1 ( ) ( ) ] VW V R V R 1 A + P V R 2 dig P db E + cd + R 2 1 A + 2 dig P 2 d + 1 A + 2 dig P ] db (7) where E denoes he gen s ceriny equivlen welh ime, where 1 1 V W nd R V 2 2 V P, nd where R V 2. Proof. See Appendix B. A suble poin which requires specil enion is h he equilibrium fee S obined bove does no induce he gen o dop he principl s opiml policy A. If his S is offered o he gen, he gen cn lwys chieve his ceriny equivlen welh E by choosing A nd cn perform beer by choosing policy A s long s c W A <c W A.s. Given he S in Equion (7), i is esy o verify h he gen s pricipion consrin is sisfied. his consrin hus drops ou of he principl s mximizion problem. In he Holmsröm Milgrom (1987) nd Schäler Sung (1993) problems, he gen s cion A does no pper in he diffusion erm, nd boh 2 nd he sock price vecor P re bsen from he conrc form. I cn be shown h he firs-order condiion (FOC) of he gen s Bellmn equion wih respec o A leds o n explici expression for 1 in erms of A. Consequenly he gen s Bellmn equion is essenilly sepred from he principl s problem, which cn hus be solved independenly. he principl s problem is hen o deermine he opiml policy A. In he Sung model, he oupu process is given by dw = Ad + db, where boh A A W nd W re one-dimensionl. he gen conrols A nd seprely, nd he principl observes eiher he whole W process or he erminl vlue W only. 13 he use of he FOC wih respec o A lone elimines 1 nd V from he expression for he opiml fee. Since 2 nd 13 Since is one-dimensionl, if he principl observes he whole W process, hen she cn infer excly. 182
Opiml Conrcs in Porfolio Mngemen Problem he sock price vecor P re bsen in his model, he opiml fee depends only upon A nd. In he presen porfolio mngemen problem, however, he FOC of he gen s Bellmn equion is no longer sufficien o provide n expression for 1 nd 2 in erms of A. herefore we mus solve for 1, 2, nd V W P explicily in order o rrive n opiml conrc S P W h implemens A. In he nex wo secions we solve for he opiml conrcs for wo ypes of problems. 4. An Opiml Conrc in Closed Form he invesor s objecive is o solve for n opiml conrc h implemens her opiml policy. Given he equilibrium fee in Equion (7) h sisfies he mnger s pricipion consrin, our sregy is o solve for A, 1, nd 2 so s o mximize he invesor s expeced uiliy. Given A, 1, nd 2, we shll hen consruc n opiml conrc in erms of P nd W from Equion (7). he poin is h he opiml conrc S P W mus no only solve he mnger s dynmic mximizion problem, bu lso reduce o Equion (7) in equilibrium when A is indeed he opiml policy. If here exiss se of soluions, hen he resuling conrc mus be opiml for our problem. 14 For rcbiliy, we ssume h he invesor s uiliy funcion is of he negive exponenil form. o solve he invesor s problem, we define her vlue funcion J W P from he S in Equion (7) s { J W P = sup E 1 exp R A 1 2 R p W E cdu p = sup A 1 2 he invesor s Bellmn equion is hen given by { J W P R 2 1 A + 2 dig P 2 du 1 A + 2 dig P ] ]} db u R p c + R p 2 R + R p 1 A + 2 dig P 2 ] + J + J W rw + A h + R p 1 A + 2 dig P ] A + 1 2 J WWA A + J P dig P + R pdig P 1 A + 2 dig P + 1 } 2 r J PP dig P dig P + J WP dig P A (8) 14 Noe h his represens he bes possible resul h he invesor cn chieve. Soluions my no lwys exis becuse he soluions for A, 1, nd 2, which re deermined from he invesor s mximizion problem, my no lwys sisfy he FOC of he mnger s Bellmn equion. 183
he Review of Finncil Sudies /v 16 n 1 23 wih he boundry condiion h J W P = 1 R p exp R p W. Wihou loss of generliy, we cn ignore he consn erm E. We now need specific cos funcion o proceed. Assume h he cos funcion is given by c A W = 1 2 A k A + W where k denoes n N N mrix, wih elemens k ij being funcions of ime, nd where is consn. Noe h his cos funcion includes 1 2 N i=1 k i A 2 i s specil cse when k is digonl mrix wih k ii = k i nd =. his specil cos funcion is widely used in sndrd morl hzrd problems where A denoes he gen s effor vecor. hough i is beyond our model, by doping his rcble cos funcion we re implicily ssuming h he more he mnger invess in he risky sses, he more effor he mus expend in cquiring informion bou he risky sses s well s in monioring he sock price movemens. he mnger s mrginl effor nd cos increse wih he level of invesmen in risky sses. 15 If k =, hen we hve c W = W, h is, he coss re proporionl o he size of he fund. his my be inerpreed s he mnger s opering coss for mnging he porfolio. A muul fund ypiclly repors n opering cos of.5% of he ol sses under mngemen nd chrges is invesors for his moun in ddiion o he mngemen fees. 16 In sum, he more he mnger invess in risky sses nd he higher he vlue of he mnger s porfolio, he higher he cos h he incurs. c A W d represens he mnger s ol coss in he ime inervl beween nd + d. he nex heorem summrizes key resul of he ricle. heorem 1. Given he bove cos funcion (or wih he ddiion of consn erm), he opiml porfolio policy nd he opiml conrc re given by A = f 1 k + f 1 = ( 1 ) e r + r r R R p R + R p f 2 1 ] 1h (9) 15 In ypicl morl hzrd problem where he gen influences he drif re or he men of n oupu, if he gen is given fixed pymen, hen he will no exer ny cosly effor. An incenive conrc is o induce he gen o expend he righ effor level. In our deleged porfolio problem, he gol of n incenive conrc is o induce he mnger o inves he righ moun in he risky sses. If he mnger is given fixed compension, hen he will inves ll he funds in he riskless sse in order o minimize coss, since invesing in he riskless sse requires no effor in informion cquisiion nd oher cive mngemen civiies. he inclusion of ime-dependen funcion k i in he cos funcion is minly for generliy nd implies h he cos for he mnger s effor ssocied wih mnging he porfolio my chnge over ime. We shll show h he form of our opiml conrcs does no depend criiclly upon he form of k i. 16 As shown in Proposiion 1, he invesor reimburses he mnger his ol coss s pr of he mnger s compension scheme. 184
Opiml Conrcs in Porfolio Mngemen Problem nd R p S P W = F + W + R +R p R +R p ] W f 1 A dig P 1 dp (1) respecively, where F denoes consn. Proof Overview. he deiled proof is given in Appendix C. In summry, we cn solve for he opiml A, 1, nd 2 from he invesor s Bellmn equion nd is hree FOCs. In he bsence of he FOC of he mnger s mximizion problem, here re muliple soluions for 1 nd 2, bu he opiml porfolio policy A is uniquely deermined. king he mnger s FOC ino ccoun, we hen find he se of soluions for 1 nd 2 which gurnees h he resuling conrc implemens he invesor s opiml policy. Given he bove opiml conrc S P W, he mnger dops he invesor s opiml porfolio policy A, nd boh he mnger nd he invesor shre he risk ssocied wih he sock prices or he Brownin moion. Given h he mnger dops A in equilibrium, he nex corollry presens he opiml fee S in differen form. Corollry 1. he opiml fee S cn be expressed s R p S = F + W R + R p ( ) R ] + W A dig P 1 dp (11) R + R p r where F denoes consn. If k = or c A W = W, hen S cn lso be doped s n opiml conrc, implemening he invesor s opiml policy A. Proof. See Appendix C. Remrk 1. he opiml conrc or fee is of symmeric form, wih he benchmrk being porfolio of risky sses. f 1 A dig P 1 in Equion (1) or A dig P 1 in Equion (11) cn be inerpreed s he number of shres invesed in he risky sses in he benchmrk porfolio. Unlike pssive index benchmrk porfolio in which he number of shres in ech risky sse is ypiclly fixed, our benchmrk porfolio invess ime-dependen nd sochsic number of shres in ech risky sse. According o he opiml conrc or fee, he invesor should py he mnger fixed fee, frcion of he ol sses under mngemen, plus bonus or penly depending upon he excess reurn beween he mnged porfolio nd he benchmrk porfolio. Our conrc no only provides possible heoreicl suppor for R 185
he Review of Finncil Sudies /v 16 n 1 23 he exising regulion h resrics he incenive fee pid o muul or pension fund o be of he symmeric form, bu lso offers n esily enforceble benchmrk gins which he fund s performnce should be mesured. Furhermore, dding consn o he cos funcion would only dd consn erm o he opiml conrc or fee. Everyhing else would remin inc. Remrk 2. his deleged porfolio mngemen problem cn lso be inerpreed s sndrd principl-gen problem in which he dw process denoes he oupu process for projec nd he gen s cion influences boh he drif nd he diffusion erms simulneously. Our model herefore represens n exension of he Holmsröm Milgrom (1987) nd Schäler Sung (1993) models in which he gen conrols he drif only. he dp vecor process my be inerpreed s ddiionl signls observble o boh he principl nd he gen. Remrk 3. he opiml porfolio policy is deerminisic funcion of ime. As Holmsröm nd Milgrom (1987) poin ou h opiml cions nd opiml conrcs in one-period principl-gen relionship re ypiclly very compliced. Our opiml conrc given in Equion (1) is quie simple nd holds regrdless of he funcionl form of k. he cos cn even be ph-dependen funcion of W, hough he opiml conrc mus be ph independen. Since he conrc depends on he ineremporl sock prices, observing he sock prices coninuously does dd vlue o he principl. Remrk 4. Recll h he equilibrium fee S represens he equilibrium moun h he invesor pys o he mnger if he mnger dops he opiml policy A. GivenA, S is equivlen o he opiml conrc S P W. S my or my no implemen A. However, when c A W = W, Corollry 1 shows h S lso implemens A. In oher words, our problem hs wo equivlen soluions S nd S P W. According o heorem 1, we cn lwys use benchmrk porfolio in he mnger s compension scheme regrdless of he cos funcion. When he cos funcion is independen of he porfolio policy vecor A, Corollry 1 presens n lernive conrc in which he cos funcion plys n imporn role. When he cos is consn, he opiml conrc reduces o liner shring rule. his is, of course, no in violion of he Amendmen, since he rule requires he incenive compension o be of symmeric form only when benchmrk is doped. When he cos funcion is proporionl o he size of he fund, he symmeric incenive performnce fee provides efficien risk shring bou he sochsic cos funcion beween he invesor nd he mnger s well s incenives for he mnger o dop he opiml porfolio policy A. Since he mnger is risk verse, he hs n incenive o follow he benchmrk porfolio A given n incenive performnce fee. If he mnger dops A, hen i cn be shown h he incenive performnce fee or he ls erm in S is equl o R p W d plus R +R p 186
Opiml Conrcs in Porfolio Mngemen Problem consn, which llows he mnger nd invesor o shre he risk ssocied wih he ph-dependen cos funcion efficienly. Since he issue regrding opiml conrcs nd pproprie benchmrks for fund mngers is of gre ineres nd hs no been ddressed in generl principl-gen frmework in he lierure, we nex presen noher opiml conrc in closed form using more generl uiliy funcions for he invesor. 5. An Addiionl Opiml Conrc in Closed Form In his secion we solve he problem using more generl uiliy funcions for he invesor. Equion (7) for he mnger s opiml fee is vlid regrdless of he invesor s preferences. When he invesor s preference is of nonexponenil form, he invesor s dynmic mximizion problem becomes nlyiclly inrcble in he presence of nonrivil cos funcion. o gin insigh ino our model, we nex solve specil cse in which he mnger s cos funcion is consn nd he mrkes re complee, h is, d = N. he purpose of his secion is o demonsre wih more exmples h he opiml conrcs re of he symmeric form nd h porfolio of risky sses cn indeed be doped s n pproprie benchmrk wihin he frmework considered in his ricle. I shll be seen h i is sill ineresing nd nonrivil o solve for robus opiml conrcs in he second-bes cse even wih consn cos funcion. We firs consider he firs-bes soluions nd hen show h he firs-bes nd he second-bes soluions coincide in equilibrium. 5.1 he opiml fee in he firs-bes cse he firs-bes soluions rise when he invesor cn force he mnger o dop ny specific porfolio policy. As resul, he mnger s incenive compibiliy consrin drops ou of he invesor s mximizion problem. he invesor s problem is hus o choose n opiml fee 17 subjec o only he mnger s pricipion consrin nd o deermine he opiml porfolio policies subjec o he budge consrin. he nex proposiion presens n expression for he opiml fee. Proposiion 2. he opiml fee S in he firs bes cse is given by S = E + 1 y y + 1 y B 2R R B (12) where he consn vecor y is defined s y 1 h. Proof. See Appendix D. Remrk 1. he S in Equion (12), which depends upon he Brownin moion B B, cnno be limiing cse 18 of n opiml conrc 17 he opiml fee is he sme s he opiml conrc in he firs-bes siuion. 18 We define he limiing cse s he one in which boh k nd go o zero. 187
he Review of Finncil Sudies /v 16 n 1 23 S P W when c W A is generl funcion of A nd W, nd is hus no robus conrc in he second-bes soluions. For exmple, when k nd go o zero, he opiml conrc in Secion 4 does no reduce o his S, which represens only he opiml fee in he secondbes cse. According o heorem 1 nd Corollry 1, when he cos funcion goes o zero, he opiml conrc reduces o eiher S P W = F + R p W + R W f R +R p R +R p 1 A dig P 1 dp or S = F + R p W. Our liner conrc S generlizes he liner risk-shring R +R p rule obined by Wilson (1968), Ross (1973, 1974), Lelnd (1978), nd ohers in one-period seing. I cn be shown h he opiml soluion A, boh conrcs re equivlen o he firs-bes fee given in Equion (12). Remrk 2. he opiml fee S in Equion (12) holds for ny invesor wih smooh uiliy funcion, becuse he invesor s uiliy funcion is no required for he derivion of Equion (12). In oher words, he mnger s bsolue moun of compension is lwys he sme regrdless of he invesor s preferences. Similr clculions cn be performed for mnger wih generl uiliy funcion. For exmple, if U = log, hen S is given by S = E exp y B B. In generl, we cn conclude h in he firs-bes siuion nd given mnger s preference, ll invesors, regrdless of he preferences, py he sme moun o he mnger. We nex derive n opiml conrc in erms of W nd P h implemens he invesor s opiml policy. 5.2 he opiml conrc in he second-bes cse In he second-bes siuion where he invesor my no force he mnger o choose specific porfolio policy, robus opiml conrc S P W my no be solved from he invesor s problem lone s in he firs-bes soluions. We mus ke he mnger s dynmic mximizion problem ino ccoun. 19 he nex heorem presens n opiml conrc h implemens he invesor s opiml policy nd h reduces o he firs-bes fee in equilibrium. heorem 2. When he invesor s uiliy funcion is given by U p W S = 1 W b S b b<1, he opiml porfolio policy is given by A = e r { e r R + 1 1 b 1 exp W e r ( E 1 1 b y B + 1 2b 2 1 b 2 y y )] y y 2R ]} y 1 19 Specificlly, given S P W, he invesor s opiml porfolio policy vecor A mus sisfy he mnger s Bellmn equion nd is FOC. In ddiion, i should be verified h he regulriy condiion given in Appendix A is sisfied. 188
Opiml Conrcs in Porfolio Mngemen Problem { e r = e r e r W + e r E 1 b R 1 b 1 y y 2R + 1 R y y + 1 R y B ]}y 1 (13) where W nd y B B or he price vecor P hve one-o-one correspondence hrough Equion (14): e r W = e r E 1 y y + 1 y y + 1 ] y B 2R R R B ( + W e r E 1 )] y y 2R 1 exp 1 b y B + 1 2b ] 2 1 b 2 y y (14) he opiml conrc h implemens he policy in Equion (13) is given by S P W = F + ne r ra 1 1 ] y 1 d + n R 1 W ] 1 + n 2 W ne r A 1R y 1 n 2 ] dig P 1 dp (15) where F denoes consn nd where n 1, n 2, nd n re rbirry posiive consns, wih n = n 1 + n 2 1. Proof Overview. he deiled proof is given in Appendix D. Conjecure h 1 y = R 1 A + 2 dig P Subsiuing he bove relion ino Equion (7) immediely yields S = E + 1 y y + 1 y B 2R R B his mens h if we cn find n opiml conrc bsed on se of soluions for 1 nd 2, hen he second-bes conrc shll reduce o he firs-bes one in equilibrium. Appendix D shows h if ] 1 1 = ne r nd 2 = y 1 ne r A dig P 1 R hen we cn rrive he opiml conrc given in Equion (15) by rewriing he S in Equion (7) in erms of W nd P. Appendix D hen 189
he Review of Finncil Sudies /v 16 n 1 23 shows h he resuling conrc implemens he invesor s opiml policy A or h given he bove S P W, A solves he mnger s dynmic mximizion problem. Remrk 1. his opiml conrc is symmeric. According o his conrc, he invesor should py he mnger fixed fee, sochsic moun involving he sock prices, frcion of he ol sses under mngemen, plus bonus or penly depending upon he porfolio s excess reurn relive o benchmrk porfolio. he pproprie benchmrk is once gin porfolio of risky sses rher hn pssive index. In his benchmrk porfolio, he number of shres invesed in ech sock vries wih ime nd he sock price. Remrk 2. I is well known Meron (1969, 1971)] h in he bsence of mnger, n invesor wih power uiliy funcion invess consn frcion of her welh in he risky socks. I hs been shown here h in he presence of mnger, he porfolio policies re quie differen: he moun invesed in he risky socks cn be more or less hn he consn frcion of her welh becuse of he normlly disribued B B erm in Equion (13). Since in he complee mrkes here is one-o-one correspondence beween he Brownin moion B nd he sock prices P, A cn lso be expressed in erms of P. 6. Conclusion his ricle provides new soluions o he conrcing problem beween n individul invesor nd professionl porfolio mnger. Vrious opiml conrcs re obined in closed form by considering boh he mnger s nd he invesor s dynmic mximizion problems nd heir firs-order condiions. Our opiml conrcs do employ benchmrk porfolio nd re indeed symmeric. hey sugges h he opiml benchmrk is porfolio of risky sses in which he number of shres invesed in ech sse vries over ime, rher hn pssive index. While our conribuion hs exended he undersnding of opiml conrcs beween invesors nd fund mngers nd seems o suppor policy of symmeric conrcs, o drw more conclusive policy implicions my require exensions of our model o more generl seings. For exmple, while our model requires he invesor o inves ll of her welh wih he mnger, new resuls my emerge when he invesor enruss only pr of her welh o he mnger nd invess he res in n index fund, which incurs relively lile cos. 2 Furher implicions my lso rise when mny invesors re llowed o inerc wih mny mngers. In ddiion, new implicions my rise when i is recognized h he muul or pension fund 2 Bizer nd DeMrzo (1999) consider opiml conrcs when gens cn sve, borrow, nd deful. 19
Opiml Conrcs in Porfolio Mngemen Problem problem involves vericlly reled principl-gen relions, h is, he relion beween individul invesors nd he fund compny nd h beween he fund compny nd he mnger who cively mnges he porfolio. 21 Appendix A: Proof of nd Verificion Resul for he Agen s Bellmn Equion Since he pril derivives of he gen s vlue funcion wih respec o he sock prices do no ply ny role in deermining opiml conrcs nd opiml policies in he presen ricle, we shll ignore hem in his ppendix for simpliciy of presenion. We firs show h A mus sisfy he Bellmn Equion Equion (6)]. By definiion, A u solves he following mximizion problem of V W = E 1 exp { R R E + + 1 rw + A h + 2 dig P c du ]}] + 1 A + 2 dig P db u (16) Muliplying boh sides of he bove equion by exp R du+ db u gives { exp R + 1 rw + A h + 2 dig P c du + 1 A + 2 dig P db u ]}]V W = E 1 exp { R R E + + 1 A + 2 dig P db ]}] + 1 rw + A h + 2 dig P c ] d Since he condiionl expecion is mringle, he drif of he process on he lef-hnd side mus vnish. Evluing is drif using Io s lemm, we find { = H W A V W R + 1 rw + A h + 2 dig P c 1 2 R2 1A + 2 dig P 2 } + V + V W rw + A h R 1 A + 2 dig P ] A + 1 2 V WW A 2 ] (17) We now show h under regulriy condiion o be specified below, Equion (17) is equivlen o he gen s Bellmn equion Equion (6)]: H W A = sup H W A H W A A 21 Gervis, Lynch, nd Muso (1999) relize h muul fund s mnger does no negoie direcly wih is invesors, bu rher wih bord of direcors which is pid o monior he fund s operion. hey develop wo-period model h finds posiive economic role for his exr lyer of relionship. 191
he Review of Finncil Sudies /v 16 n 1 23 where A is n rbirry policy for he iniil welh W. Le W A be he ssocied welh process solving dw A = rw + A h d + A db Suppose here exiss porfolio policy A such h H W A >H W A = for ll, we cn hen define sochsic process J A s J A { = exp R du c du+ ]} 1 dw A u + 2 dp u V W A where V W A is well-defined vlue funcion 22 wih he boundry condiion of V, W A = 1 exp R R E, nd where u W A P wih he excepion h c c u W A. We ssume h in V W A, A s = A s, s>, mening h he conrol A s in he process J A is swiched o he opiml conrol A s immediely fer ime. A he erminl de, J A is hen given by J A = 1 ]} exp { R R E + d c d+ 1 dw A + 2 dp Similrly, sochsic process J A my be defined for n rbirry policy A, including A. Le J A A J for ll A, implying h his sochsic process J srs he sme poin ime nd my end differen poins ime, depending upon he porfolio policy ken beween imes nd. A srighforwrd pplicion of Io s lemm yields ] dj A = F A H W A d+ W A db where F A F A nd W A re given by { = exp R du c du+ 1 dw A u + ]} 2 dp u nd W A = V W W A A R V W A 1 A + 2 dig P ] respecively. We hus hve J A = J + F A H W A d+ F A W A db >J + F A W A db (18) where we hve used he fc h boh F A nd H W A re posiive for ll. Since he gen s uiliy funcion is of negive exponenil form, he vlue funcion V W A or J A cnno be posiive. Rerrnging he bove equion, we hve J A < J F A W A db Following n rgumen in Duffie (1996, Chper 9): since J A is nonnegive, posiive process M cn be defined by M = J u W A db u. We know h M is locl F A u 22 I is coninuously differenible in nd wice coninuously differenible in W A. 192
Opiml Conrcs in Porfolio Mngemen Problem mringle nd h posiive (or bounded from below) locl mringle is super-mringle. By king he expecions of ech side of nd rerrnging he bove equion, we obin E J A = E 1 exp { R R E + d c d + 1 dw A + 2 dp ]]}] >J (19) A similr clculion pplies wih A = A, yielding Assuming h E F A J A = J + F A W A db W A 2 d <, we know h E F A W A db =. see, e.g., Duffie (1996) nd Proer (199)]. We cn hen ke he expecion of boh sides nd hve E ] J A = E 1 exp { R R E + d c d + 1 dw A + 2 dp ]]}] = J (2) where W A P. Equions (19) nd (2) conrdic he ssumpion h A u solves he problem in Equion (16). Hence H W A cnno be sricly posiive for every. Furhermore, i cn be shown h H W A my no be sricly posiive ny single. Suppose H W A > for ime. We cn hen consruc new policy A, which selecs A beween nd +d nd he opiml policy A ny oher ime. Equion (18) would sill hold for J A, which would led o E J A >E J A, conrdicing gin he ssumpion h A is he opiml policy. herefore we cn conclude h = H W A H W A for ll, compleing he proof h he Bellmn equion is necessry condiion for A o be n opiml policy. We now show h A, which sisfies he Bellmn equion, is indeed n opiml policy. he proof is essenilly he sme s discussed bove. For n rbirry policy A, he Bellmn equion ses h H W A. Similrly we cn define process J A s { J A = exp R du c du+ ]} 1 dw A u + 2 dp u V W A wih he sme boundry condiion h V W A = 1 R exp R E for ll A. Agin, we ssume h in V W A, A s = A s s>. 23 From he Bellmn equion nd Io s lemm we hve J A J + F A W A db where he equliy holds A = A. Here he super-mringle rgumen does no pply. Insed, we need o ssume h E F A W A 2 d <. We cn hen ke he expecions nd rrive E J A = J E J A, proving h A is n opiml policy. 23 Schäler nd Sung (1993) define differen process in which A s coincides wih A s prior o nd wo conrols differ ferwrds. In our definiion of J A, A s differs from A s prior o nd including nd is idenicl o A immediely fer. 193
he Review of Finncil Sudies /v 16 n 1 23 Appendix B: Proof of Proposiion 1 By Io s lemm, dv W P is given by { dv W P = V + V W rw + A h + 1 2 V WWA A + V dig P + V dig P P WP A + 1 } 2 r V PP dig P dig P d + V W A + V dig P db P From he Bellmn equion Equion (6)], we hve V + V W rw + A h + 1 2 V WWA A + V P dig P + V WP dig P A + 1 2 r V PP dig P dig P = V W P R c + 1 rw + A h + 2 dig P 1 2 R2 1 A + 2 dig P 2 + V W R 1 A + 2 dig P A + V P R 1 dig P A + dig P dig P 2 where 2. Combining he bove wo equions yields { dv W P = V W P R c + 1 rw + A h + 2 dig P 1 2 R 1 A + 2 dig P 2 } + V W R 1 A + 2 dig P A + V P R 1 dig P A + dig P dig P 2 ] d + V W A + V P dig P db (21) A = A. Following Holmsröm nd Milgrom (1987) nd Schäler nd Sung (1993), we define n E process s 24 R E = log R V W P Using Equion (21) for dv W P, we obin R de = dv V + 1 ( ) 2 dv = dv 2 V V + 1 ( ) 2 VW A Ad 2 V = R c + 1 rw + A h + 2 dig P d + 1 2 R2 1A + 2 dig P 2 d V W V R 1 A + 2 dig P Ad V P V R 1 dig P A + dig P dig P d 2 VW V A + V ] P V dig P db + 1 V W 2 V A + V 2 P V dig P d 24 his definiion is merely mhemicl mnipulion nd hs no economic inerpreion in he curren model. E my be ny process h srs E nd ends E. 194
Opiml Conrcs in Porfolio Mngemen Problem Noe h + 1 rw + A h + 2 dig P d = d+ 1 dw + 2 dp 1 A + 2 dig P db We hus hve R de = R cd R d + 1 dw + 2 dp + R 1 A + 2 dig P db + 1 2 R2 1A + 2 dig P 2 d V W V R 1 A + 2 dig P Ad V P V R 1 dig P A + dig P dig P d 2 + 1 V W 2 V A + V 2 P V dig P VW d V A + V ] P V dig P db = R cd R d + 1 dw + 2 dp + 1 2 R2 1A + 2 dig P 2 d V W V R 1 A + 2 dig P Ad V P V R dig P 1 A + 2 dig P d + 1 V W 2 V A + V 2 P V dig P d VW V A R 1 A + V ] P V dig P R 2 dig P db = R cd R d + 1 dw + 2 dp + 1 2 R2 1A + 2 dig P 2 d VW R V A + V ] P V dig P 1 A + 2 dig P d + 1 V W 2 V A + V 2 P V dig P d ( ) ( ) ] VW V V R 1 A P + V R 2 dig P db = R d + 1 dw + 2 dp ( ) ( ) VW V 2 A P + dig P d + 1 2 V R 1 V R 2 ( ) VW + R cd V R 1 A + A = A. Inegring beween nd,wege ( V P V R 2 ) dig P ] db S = E + cd+ 1 ( ) ( ) VW V 2 2R V R 1 A P + V R 2 dig P d 1 ( ) ( ) ] VW V R V R 1 A P + V R 2 dig P db E + cd+ R 2 1 A + 2 dig P 2 d + 1 A + 2 dig P db 195
he Review of Finncil Sudies /v 16 n 1 23 Remrks. Noe h he vlue funcion V W P does no represen he gen s expeced uiliy funcion, which is given by V W P = sup E 1 exp { R A R S = V W P exp { R du+ ]} ] c d 1 dw u + 2 dp u ]} c du Since V W P is no uiliy funcion, i my no be incresing in W. Of course, he expeced uiliy funcion mus be incresing in W. Given n S P W in Equion (2), he definiion of V W P my no be unique. For insnce, if we define n E process s E = W, S P W my be expressed in wo equivlen forms: S P W = W + = W + + W P d + dw + 2 W P dp 1 W P dw + W P d + 1 W P dw 2 W P dp Accordingly, wo differen vlue funcions cn be defined s V 1 W P = sup E 1 exp { R A u R W + u W P du c u W A du ]} ] + 1 u W P dw u + 2 u W P dp u nd V 2 W P = sup E 1 exp { R A u R W + dw u + u W P du ]} ] c u W A du + 1 u W P dw u + 2 u W P dp u = 1 R exp R W W V 1 W P where V 1 W P = V 2 W P only =. Bu he wo vlue funcions correspond o he sme conrc S P W nd he expeced uiliy funcion is unique. In ddiion, he opiml policy A remins he sme irrespecive of he definiion of he vlue funcion. A sligh exension of Proposiion 1 leds o he following corollry. Corollry B. If he welh nd he sock price processes ke more generl forms such s 25 dw = f W A d + g W A db f W d+ g W db nd dp = f P d+ g P db 25 We ssume h f nd g sisfy he necessry regulriy condiions nd h hey represen he relevn vecors or mrices. 196
Opiml Conrcs in Porfolio Mngemen Problem hen he expression for he equilibrium fee S is given by S = E + cd + 1 V W 2R V g W R 1 g W + V 2 P V g P R 2 g P d 1 VW R V g W R 1 g W + V ] P V g P R 2 g P db E + cd + R 2 1 g W + 2 g P 2 d + 1 g W + 2 g P db (22) he proof for his corollry is excly he sme s for Proposiion 1 nd is hus omied. his represenion of S exends hose in Holmsröm nd Milgrom (1987) nd Schäler nd Sung (1993), where g W A is independen of A nd where he sock price processes re bsen from he conrc spce. Appendix C: Proofs of heorem 1 nd Corollry 1 Proof of heorem 1. Recll from Secion 4 h he invesor s Bellmn equion is given by { = sup J W P R p c + R p A 1 2 2 R + R p 2] 1 A + 2 dig P + J + J W rw + A h + R p 1 A + 2 dig P A + 1 2 J WWA A + J dig P + R P p dig P 1 A + 2 dig P + 1 } 2 r J PP dig P dig P + J dig P WP A (23) wih boundry condiion h J W P = 1 Rp exp R pw. he FOCs of he invesor s Bellmn equion wih respec o 1, 2, nd A yield nd J W P R + R p A 1 A + 2 dig P + J W A A + J P dig P A = J W P R + R p dig P 1 A + 2 dig P + J W dig P A + dig P dig P J P = J W P R p c A + R + R p 1 1 A + 2 dig P + J W h + 2R p 1 A + R p 2 dig P + J WW A + R p 1 dig P J P + dig P J WP = respecively. Conjecure h J W P kes he following form: J W P = 1 R p exp R p f 1 W + f 2 P + f 3 wih boundry condiions h f 1 = 1 nd f 2 = f 3 =. Subsiuing J W P ino nd simplifying he bove FOCs, we obin R + R p A 1 A + 2 dig P + f 1 R p A A + R p A f 2 dig P = 197
he Review of Finncil Sudies /v 16 n 1 23 nd R + R p dig P 1 A + 2 dig P + f 1 R p dig P A + R p dig P f 2 dig P = c A R +R p 1 1 A + 2 dig P +f 1 h+2r p 1 A+R p 2 dig P R p f 2 1 A+R p 1 f 2 dig P R p f 1 f 2 dig P = In order o rrive se of soluions h sisfy hese hree FOCs s well s he invesor s Bellmn equion, i is necessry h f 2 =. In ddiion, if 1 A + 2 dig P = R p R + R p f 1 A (24) hen he FOCs wih respec o 1 nd 2 re sisfied. Subsiuing Equion (24) ino he FOC wih respec o A, wehve = c A + f 1 h 1 R p f 1 A + f 1 2R p 1 A + f 1 R p 2 dig P R p f 2 1 A = c A + f 1 h + R p f 1 1 A + R p f 1 2 dig P R p f 2 1 A = c A + f 1 h + R p f 1 1 A + 2 dig P R p f 2 1 A R p = c A + f 1 h + R p f 1 f R + R 1 A R p f 2 1 A p = c A + f 1 h f 2 R R p 1 A R + R p he opiml policy vecor is hus given by A = f 1 k + R ] R p f 2 1 1h R + R (25) p Here he A vecor is uniquely deermined, independen of he soluions for 1 nd 2. Subsiuing J W P ino nd simplifying he invesor s Bellmn equion, we hve 1 = 2 A k A + W + R ] p 2 f 1 A A + f 1 rw + A h + R ] p f R + R 1 A A p 1 2 R pf 2 1 A A + f + f 3 1 W (26) o elimine he W erms in he bove equion we mus hve + rf 1 + f 1 = f 1 = 1 We hus obin h ( f 1 = 1 ) e r + r r Noe h A is independen of W, nd s resul, proper choice of f 3 will hen ensure he sisfcion of he invesor s Bellmn equion. 198
Opiml Conrcs in Porfolio Mngemen Problem We now discuss he soluions for 1 nd 2. For he opiml conrc o be boh ph independen nd implemenble, we rrive se of soluions ( 1 = f 1 = 1 ) e r + r r nd 2 = R f R + R 1 A dig P 1 p Consrucing n opiml conrc from Equion (7), we rrive S P W = consn + = consn + W d+ W d+ 1 A + 2 dig P db ( 1 ) e r + ] dw rw d r r + 2 dp F +W + 2 dp = F + R p W + R ] W f R +R p R +R 1 A dig P 1 dp p which is Equion (1) in heorem 1. In he bove derivion we hve used he fc h A nd 2 P re deerminisic funcions of only. We now verify h given his S P W, he mnger s Bellmn equion nd is FOC re sisfied. he mnger s Bellmn equion is given by { = sup V R 2 dig P 1 A 2 A k A W 1 } 2 R 2 dig P 2 + V + V W rw + A h R 2 dig P A + 1 ] 2 V WWA A where we hve omied he V, V P PP, nd V WP erms becuse hey do no ply ny role in he deerminion of he mnger s opiml policy A. Conjecure h he mnger s vlue funcion is given by V W = 1 R exp R b 1 W + b 2 which is independen of P. Here b 1 nd b 2 re funcions independen of W wih boundry condiions h b 1 = 1 nd b 2 =. Subsiuing V W ino nd simplifying he mnger s Bellmn equion, we hve = sup 2 dig P 1 A 2 A k A W 1 2 R 2 dig P 2 + b W + 1 b 2 + b 1 rw + A h R 2 dig P A 1 ] 2 R b 2 1 A A From his equion, i cn be shown h ( b 1 = 1 ) e r + r r f 1 As long s he mnger s opiml porfolio policy is deerminisic, his Bellmn equion will be sisfied wih proper choice of b 2. 199
he Review of Finncil Sudies /v 16 n 1 23 We nex show h he invesor s opiml policy A sisfies he FOC of he mnger s Bellmn equion. he FOC yields k A + b 1 h R b 1 2 dig P R b 2 1 A = k A + f 1 h + If A = A, hen we hve R2 f 2 R + R 1 A R f 2 1 A p k A + f 1 h f 2 1 R R p R + R p A = he mnger s FOC is hus sisfied by A. I cn be shown h he regulriy condiion given in Appendix A is sisfied. 26 herefore S P W nd A form n opiml soluion o he problem. Proof of Corollry 1. Given he opiml policy A, wehve dw = r W A 1 d+ A dig P 1 dp or A dig P 1 dp = dw r W A 1 d I cn be shown h f 1 A dig P 1 dp = consn + f 1 dw rw d ( 1 r ) W + r = consn + dw rw d ( = consn + 1 ) W + A dig P 1 dp r r Subsiuing he bove relion ino Equion (1) immediely yields ( ) ] ( ) R S = F p R R + + W A dig P 1 dp R + R p R + R p r R + R p r = F + R ( ) p R ] W + W A dig P 1 dp R + R p R + R p r Noice h he opiml fee S my or my no implemen he invesor s opiml policy A. 26 Noice h W A s defined in Appendix A is given by ( ) ] W A VW = R V W R V 1 A 2 dig P = R V W 1 A + 2 dig P = R R p f R + R 1 A V W p From he definiion of F A s given in Appendix A, i cn be shown h F A V W = 1 R exp R S c d. Since f 1 nd A re boh bounded, we cn show h E A F W A 2 d ] = A E F W A 2 d <, using he relion 1 A + 2 dig P = R p R+Rp f 1 A nd he expression for S, which coincides wih S P W in equilibrium. Here, he use of Fubini s heorem is jusified becuse he inegrnd is posiive. 2
Opiml Conrcs in Porfolio Mngemen Problem We now show h when k = orc A W = W, he bove S implemens A. Given his S, he mnger defines vlue funcion s V W P = 1 R p E R exp { R + R + R p ( R R + R p ( R ) ] W W u du r }] R + R p ) A u dig P 1 dp u r wih he boundry condiion h V W P = 1 exp R R + R+Rp R+Rp r W. he mnger s Bellmn equion kes he sme form s in he proof of heorem 1 excep h 2 nd b 1 re now given by ) R 2 = ( R + R p r A dig P 1 b 1 = R ( p 1 ) e r + R + R p r r he presen form for b 1 is chosen so s o elimine he W erms in he mnger s Bellmn equion s well s o sisfy he boundry condiion h b 1 = R p + R. R+Rp R+Rp r When k =, he FOC of he mnger s Bellmn equion yields h R 2 dig P R b 1 A ( ) ( R 2 R R = h + R + R p r A p 1 ) e r + ] R + R p r r R A If A = A, hen we hve h R ( R p 1 ) e r + ] A = h R R p f R + R p r r R + R 1 A = p which is he sme s he FOC of he invesor s Bellmn equion. he res of he proof follows from h of heorem 1. Rp R Appendix D: Proofs of Proposiion 2 nd heorem 2 In his ppendix we firs solve for he firs-bes fee nd hen show h he firs-bes resuls cn be implemened using he conrc form given in heorem 2. Proof of Proposiion 2. Since he mrkes re dynmiclly complee by ssumpion, we employ he mringle represenion pproch developed by Cox nd Hung (1989, 1991), Krzs, Lehoczky, nd Shreve (1987), nd Plisk (1986). 27 In his pproch, one solves dynmic porfolio problem by sepring i ino wo prs. Firs, one rnsforms he dynmic porfolio problem ino sic uiliy mximizion problem nd solves he sic problem o find he opiml welh process. hen one pplies he mringle represenion heorems o deermine he porfolio policies needed o genere he opiml welh process. he invesor s problem in he firs-bes siuion is o sup E U p W S (27) S W 27 For mringle represenion pproch wih incomplee mrkes, see, for exmple, He nd Person (1991) nd Krzs e l. (1991). 21
he Review of Finncil Sudies /v 16 n 1 23 s.. s.. E 1 ] exp R R S 1 exp R R E E W e r = E W e r W where boh consrins bind he opiml soluions S nd W, 28 nd where, wihou loss of generliy, he consn cos funcion hs been omied. Here, W insed of A ppers in he invesor s mximizion problem, becuse in he mringle represenion pproch, choosing he opiml erminl welh W is equivlen o choosing he opiml porfolio policies. denoes he equivlen mringle mesure or he risk-neurl probbiliy, 29 nd d /dp is given by = exp 1 ] 2 y y y B B where P is he originl mesure nd where y 1 h. Assume h he invesor s uiliy funcion U p is smooh everywhere wih respec o boh W nd S. 3 Forming he Lgrngin, we hve sup E U p W S ] S exp R S W R S W e r W where S nd W denoe he Lgrngin mulipliers for he consrins wih respec o S nd W, respecively. he FOCs for his poin-wise mximizion problem wih respec o S nd W re given by nd U p W S = S exp R S (28) U p W S = W e r (29) respecively. 31 he opiml fee S in erms of B B is n immedie resul of he combinion of hese wo FOCs: S exp R S = W e r (3) Equion (28) deermines S in erms of W or vice vers. Wih he id of Equion (3), Equion (28) gives W in erms of B B, from which he invesor s opiml porfolio policies re deermined. From Equion (3), we hve S = 1 ] W log e r + 1 y y + 1 y B R S 2R R B = E + 1 y y + 1 y B 2R R B where we hve used he mnger s pricipion consrin o eque he firs equion wih he second one, h is, E = 1 log W e r. R S 28 For ese of exposiion, S nd W rher hn S nd W re being used o denoe he respecive opiml soluions. 29 See Hrrison nd Kreps (1979) for he former nd Cox nd Ross (1976) for he ler. 3 A funcion is sid o be smooh if i is coninuous nd hs coninuous firs-order derivive. 31 Noe h W nd S re wo independen conrol vribles in he firs-bes soluions. I cn be shown h if U p W S is concve wih respec o W S, hen i is concve wih respec o boh W nd S. herefore he FOCs re boh necessry nd sufficien. 22
Opiml Conrcs in Porfolio Mngemen Problem Proof of heorem 2. Our objecive is o solve for conrc S P W h boh implemens he invesor s firs-bes porfolio policies nd reduces o he firs-bes fee S. Consequenly he second-bes resuls coincide wih he firs-bes ones. he firs-bes fee S is given by S = E + 1 2R y y + 1 R y B B (31) Since his S sisfies he mnger s pricipion consrin, he firs-bes opiml porfolio policies my be solved from he invesor s mximizion problem, nmely, ] 1 sup E W S b W b (32) s.. E W e r = E W e r W where b<1. As in he proof of Proposiion 2, he equivlen mringle mesure is uniquely represened by d /dp = exp 1 2 y y y B B. According o he Girsnov heorem, B B + y is sndrd Brownin moion under see, e.g., Krzs nd Shreve (1991)]. Solving for he FOC of he poin-wise sic mximizion problem in Equion (32) yields he opiml erminl welh W : W S b 1 = e r exp 1 ] 2 y y y B B (33) where is he Lgrngin muliplier. Using he fc h he discouned opiml welh process, W e r, is mringle under, h is, W e r = E W e r, we obin or e r W = E Q e r W = e r E 1 { e r d e r W = R = From Equion (34), we hve y y + 1 y y + 1 ] y B 2R R R B + b 1 1 b exp 1 b r + b 2 1 b 2 y y + 1 1 b y B + 1 2b 2 1 b 2 y y + 1 b 1 b 1 b exp 1 b r + b 2 1 b 2 y y + 1 1 b y B + 1 2b 2 1 b 2 y y ] (34) y 1 hd + db { 1 1 b e r W + 1 e r 1 R 1 b er E 1 y y + 1 y y + 1 ]} y B 2R R R y 1 hd + db (35) ] ( b 1 1 b exp 1 b r + b 2 1 b 2 y y = W e r E 1 ) y y 2R o obin A, we recll h he dw process is given by dw = rw + A h d + A db ]} 23
he Review of Finncil Sudies /v 16 n 1 23 Applying Io s lemm, we obin h d e r W = e r A hd + db (36) he unique decomposiion of d e r W nd comprison of Equion (35) wih Equion (36) led o { ( e r A = e r + W e r E R 1 )] y y 2R 1 exp 1 b y B + 1 2b ]} 2 1 b 2 y y y 1 { 1 = e r 1 b W + 1 e r R 1 1 b e r E 1 y y + 1 y y + 1 ]} y B 2R R R y 1 (37) where he equilibrium W nd y B B hve one-o-one correspondence hrough Equion (34). We cn lso express A in erms of he price vecor P. We now consruc conrc S P W h implemens he bove A. Compring he conrcs given in Equions (31) nd (7), we conjecure h 1 y = R 1 A + 2 dig P y = 1 h (38) his is, of course, jus one of mny soluions. Bu s long s he finl conrc implemens he invesor s A, his soluion will be righ one. As in he proof of heorem 1, he opiml soluion for 1 is given in his cse by 1 = ne r, wih n being consn. We hen hve ] ] 1 1 2 = y ne r A dig P 1 = y 1 ne r A dig P 1 R R (39) Subsiuing he soluions for 1 nd 2 ino Equion (7), we rrive S P W = E + 1 2R y y + ne r dw rw d + F = F + ne r A h + 2 dig P d 2 dp ne r A h + 2 dig P d + nw ne r A 1 R y 1 ] dig P 1 dp ne r ra 1 1 ] y 1 d + n R 1 W +n 2 W 1 ne r A 1 ] y ] dig P 1 1 dp n 2 R 1 where n 1 + n 2 = n. We ke n o be less hn or equl o 1. Here, ne r A n 2 1 R y 1 dig P 1 cn be inerpreed s he number of shres invesed in risky sses in benchmrk porfolio. 24
Opiml Conrcs in Porfolio Mngemen Problem We now verify h he bove conrc S P W implemens A. Given his conrc, he mnger s Bellmn equion is given by = sup V R 2 dig P 1 ] A 2 R 2 dig P 2 ne r A h 2 dig P + V + V W rw + A h R 2 dig P A + 1 ] 2 V WWA A where we hve omied he V P, V PP, nd V WP erms becuse hey do no ply ny role in deermining he mnger s opiml policy. Conjecure h he mnger s vlue funcion is given by V W = 1 R exp R b 1 W + b 2 wih he boundry condiions h b 1 = n nd b 2 =, we hen hve = sup 1 A 2 R 2 dig P 2 ne r A h + b W + 1 b 2 + b 1 rw + A h R 2 dig P A 1 ] 2 R b 2 1 A A From his equion, we obin b 1 = ne r = 1. 32 he gen s FOC is hen given by Using b 1 = 1,wehve h R 2 dig P R b 1 A = h=r 1 A + 2 dig P or 1 R y = 1 A + 2 dig P By consrucion, he invesor s opiml policy A sisfies he bove mnger s FOC. We now verify h he mnger s Bellmn equion is sisfied s well, which is no rivil becuse A is funcion of P. 33 Simplifying he mnger s Bellmn equion, we hve 1 2 R 2 dig P 2 + b 2 b 1 R 2 dig P A 1 2 R b 2 1 A A = 1 2 R 2 dig P 2 + 2 2 dig P 1 A + 1 A 2] + b 2 = 1 2 R 1 1 A + 2 dig P 2 + b 2 = y y + b 2 = 2R which is indeed sisfied wih choice of b 2 given by b 2 = 1 2R y y. herefore boh he mnger s Bellmn equion nd is FOC re sisfied. I cn be shown h he regulriy condiion given in Appendix A is sisfied. 34 32 his soluion of b 1 elimines he W erms from he mnger s Bellmn equion. 33 In heorem 1, A is deerminisic funcion of, he mnger s Bellmn equion is lwys sisfied wih proper choice of b 2. 34 Using 1 A + 2 dig P = 1 R y, we rrive W A = y V W nd F A W A = y 1 R exp R S c d. he res of he proof is srighforwrd. 25
he Review of Finncil Sudies /v 16 n 1 23 References Ackermnn, C., R. McEnlly, nd D. Rvenscrf, 2, he Performnce of Hedge Funds: Risk, Reurn nd Incenives, Journl of Finnce, 54, 833 874. Admi, A. R., nd P. Pfleiderer, 1997, Does I All Add Up? Benchmrks nd he Compension of Acive Porfolio Mngers, Journl of Business, 7, 323 35. Allen, F., 199, he Mrke for Informion nd he Origin of Finncil Inermediion, Journl of Finncil Inermediion, 1, 3 3. Allen, F., 21, Do Finncil Insiuions Mer? Journl of Finnce, 56, 1165 1175. Allen, F., nd A. M. Snomero, 1997, he heory of Finncil Inermediion, Journl of Bnking nd Finnce, 21, 1461 1485. Bhchry, S., nd P. Pfleiderer, 1985, Deleged Porfolio Mngemen, Journl of Economic heory, 36, 163 187. Bizer, D., nd P. DeMrzo, 1999, Opiml Incenive Conrcs When Agens Cn Sve, Borrow, nd Deful, Journl of Finncil Inermediion, 8, 241 269. Bolon, P., nd C. Hrris, 1997, he Coninuous-ime Principl-Agen Problems: Firs-Bes Risk Shring Conrcs nd heir Decenrlizion, working pper, Cmbridge Universiy nd Princeon Universiy. Brennn, M. J., 1993, Agency nd Asse Pricing, working pper, Universiy of Cliforni, Los Angeles. Brown, K. C., W. V. Hrlow, nd L.. Srks, 1996, Of ournmens nd empions: An Anlysis of Mngeril Incenives in he Muul Fund Indusry, Journl of Finnce, 51, 85 11. Brown, S. J., nd W. N. Goezmnn, 1995, Performnce Persisence, Journl of Finnce, 5, 679 698. Brown, S. J., W. N. Goezmnn, nd J. Prk, 1997, Condiions for Survivl: Chnging Risk nd he Performnce of Hedge Fund Mngers nd CAs, working pper, New York Universiy nd Yle Universiy. Crhr, M. M., 1997, On Persisence in Muul Fund Performnce, Journl of Finnce, 52, 57 82. Crpener, J. I., 2, Does Opion Compension Increse Mngeril Risk Appeie? Journl of Finnce, 55, 2311 2331. Chen, H. L., nd G. Penncchi, 2, Does Prior Performnce Affec Muul Fund s Choice of Risk? heory nd Furher Empiricl Evidence, working pper, Universiy of Illinois Chicgo nd Urbn-Chmpign. Chevlier, J., nd G. Ellison, 1999, Creer Concerns of Muul Fund Mngers, Qurerly Journl of Economics, 114, 389 41. Cohen, S. I., nd L.. Srks, 1988, Esimion Risk nd Incenive Conrcs for Porfolio Mngers, Mngemen Science, 34, 167 179. Core, J., nd J. Qin, 21, Opion-like Conrcs for Projec Selecion nd Producion, working pper, Boson College. Cox, J., nd C.-F. Hung, 1989, Opiml Consumpion nd Porfolio Policies When Asses Follow Diffusion Process, Journl of Economic heory, 49, 33 83. Cox, J., nd C.-F. Hung, 1991, A Vriionl Problem Arising in Finncil Economics wih n Applicion o Porfolio urnpike heorem, Journl of Mhemicl Economics, 2, 465 488. Cox, J., nd S. A. Ross, 1976, he Vluion of Opions for Alernive Sochsic Processes, Journl of Finncil Economics, 3, 145 166. Cuoco, D., nd R. Kniel, 2, Generl Equilibrium Implicions of Fund Mngers Compension Fees, working pper, Universiy of Pennsylvni nd Universiy of exs Ausin. Ds, S., nd R. Sundrm, 1998, On he Regulion of Muul Fund Fee Srucure, working pper, Hrvrd Universiy nd New York Universiy. 26
Opiml Conrcs in Porfolio Mngemen Problem Deemple, J., S. Govindrj, nd M. Loewensein, 21, Opiml Conrcs nd Ineremporl Incenives wih Hidden Acions, working pper, Boson Universiy. Dow, J., nd G. Goron, 1997, Noise rding, Deleged Porfolio Mngemen, nd Economic Welfre, Journl of Poliicl Economy, 15, 124 15. Duffie, D., 1996, Dynmic Asse Pricing heory, Princeon Universiy Press, Princeon, NJ. Dybvig, P., H. K. Frnsworh, nd J. N. Crpener, 2, Porfolio Performnce nd Agency, working pper, Wshingon Universiy in Sin Louis. Dybvig, P., nd C. Sp, 1986, Agency nd he Mrke for Muul Fund Mngers: he Principle of Preference Similriy, working pper, Crnegie-Mellon Universiy nd Wshingon Universiy in Sin Louis. Elon, E. J., M. Gruber, S. Ds, nd M. Hlvk, 1993, Efficiency wih Cosly Informion: A Reinerpreion of Evidence from Mnged Porfolios, Review of Finncil Sudies, 6, 1 22. Fung, W., nd D. Hsieh, 1997, Empiricl Chrcerisics of Dynmic rding Sregies: he Cse of Hedge Funds, Review of Finncil Sudies, 1, 275 32. Grci, D., 2, Incomplee Conrcs in Invesmen Models, working pper, Drmouh College. Grci, D., 2b, Opiml Menus of Liner Conrcs in Porfolio Selecion Models, working pper, Drmouh College. Gervis, S., A. W. Lynch, nd D. K. Muso, 1999, Deleged Monioring of Fund Mngers: An Economic Rionle, working pper, Universiy of Pennsylvni. Goezmnn, W., J. Ingersoll, nd S. A. Ross, 1997, High Wer Mrks, working pper, Yle Universiy. Golec, J. H., 1992, Empiricl ess of Principl-Agen Model of he Invesmen Advisor Relion, Journl of Finncil nd Quniive Anlysis, 27, 91 95. Gompers, P. A., nd A. Merick, 1998, How Are Lrge Insiuions Differen from Oher Invesors? Why Do hese Differences Mer for Equiy Prices nd Reurns? working pper, Hrvrd Universiy. Grinbl, M., nd S. imn, 1989, Adverse Risk Incenives nd he Design of Performnce-Bsed Conrcs, Mngemen Science, 35, 87 822. Grinbl, M., nd S. imn, 1993, Performnce Mesuremen Wihou Benchmrks: An Exminion of Muul Fund Reurns, Journl of Business, 66, 47 68. Gruber, M. J., 1996, Anoher Puzzle: he Growh in Acively Mnged Muul Funds, Journl of Finnce, 51, 783 81. Hrrison, M., nd D. Kreps, 1979, Mringles nd Arbirge in Muliperiod Securiies Mrkes, Journl of Economic heory, 2, 381 48. He, H., nd N. Person, 1991, Consumpion nd Porfolio Policies wih Incomplee Mrkes, Journl of Economic heory, 54, 259 35. Heinkel, R., nd N. M. Soughon, 1994, he Dynmics of Porfolio Mngemen Conrcs, Review of Finncil Sudies, 7, 351 387. Holmsröm, B., nd P. Milgrom, 1987, Aggregion nd Lineriy in he Provision of Ineremporl Incenives, Economeric, 55, 33 328. Ingersoll, J. E., 1987, heory of Finncil Decision Mking, Rowmn & Lilefield, NJ. Jing, W., 2, Incenives for Money Mngers Under Endogenous Risk Choice, working pper, Universiy of Chicgo. Krzs, I., J. Lehoczky, nd S. Shreve, 1987, Opiml Porfolio nd Consumpion Decision for Smll Invesor on Finie Horizon, SIAM Journl of Conrol nd Opimizion, 25, 1157 1186. 27
he Review of Finncil Sudies /v 16 n 1 23 Krzs, I., J. Lehoczky, S. Shreve, nd G.-L. Xu, 1991, Mringle nd Duliy Mehods for Uiliy Mximizion in Incomplee Mrkes, SIAM Journl of Conrol nd Opimizion, 29, 72 73. Krzs, I., nd S. Shreve, 1991, Brownin Moion nd Sochsic Clculus, Springer-Verlg, New York. Kihlsrom, R., 1988, Opiml Conrcs for Securiy Anlyss nd Porfolio Mngers, Sudies in Bnking nd Finnce, 5, 291 325. Lkonishok, J., A. Shleifer, nd R. W. Vishny, 1992, he Srucure nd Performnce of he Money Mngemen Indusry, Brookings Ppers on Economic Aciviy: Microeconomic, 24, 339 391. Lelnd, H., 1978, Opiml Risk Shring nd he Lesing of Nurl Resources, wih Applicion o Oil nd Gs Lesing on he OCS, Qurerly Journl of Economics, 92, 413 437. Lynch, A., nd D. Muso, 1997, Undersnding Fee Srucures in he Asse Mngemen Business, working pper, New York Universiy nd Universiy of Pennsylvni. Mmysky, H., nd M. Spiegel, 21, A heory of Muul Funds: Opiml Fund Objecives nd Indusry Orgnizion, working pper, Yle Universiy. Meron, R., 1969, Lifeime Porfolio Selecion Under Unceriny: he Coninuous ime Cse, Review of Economics nd Sisics, 51, 247 257. Meron, R., 1971, Opimum Consumpion nd Porfolio Rules in Coninuous ime Model, Journl of Economic heory, 3, 373 413. Plisk, S., 1986, A Sochsic Clculus Model of Coninuous rding: Opiml Porfolios, Mhemics of Operions Reserch, 11, 371 382. Proer, P., 199, Sochsic Inegrion nd Differenil Equions, Springer-Verlg, New York. Ross, S. A., 1973, he Economic heory of Agency: he Principl s Problem, Americn Economic Review, 63, 134 139. Ross, S. A., 1974, On he Economic heory of Agency nd he Principle of Similriy, in M. Blch, D. McFdden nd Shih-Yen Wu (eds.), Essys on Economic Behvior Under Unceriny, Norh-Hollnd, Amserdm. Schäler, H., nd J. Sung, 1993, he Firs-Order Approch o he Coninuous-ime Principl-Agen Problem wih Exponenil Uiliy, Journl of Economic heory, 61, 331 371. Srks, L.., 1987, Performnce Incenive Fees: An Agency heoreic Approch, Journl of Finncil nd Quniive Anlysis, 22, 17 32. Soughon, N., 1993, Morl Hzrd nd he Porfolio Mngemen Problem, Journl of Finnce, 48, 29 228. Sung, J., 1995, Lineriy wih Projec Selecion nd Conrollble Diffusion Re in Coninuous-ime Principl-Agen Problems, RAND Journl of Economics, 26, 72 743. Sung, J., 2, Opiml Conrcs Under Morl Hzrd nd Adverse Selecion: A Coninuous-ime Approch, working pper, Universiy of Illinois Chicgo. Wermers, R., 2, Muul Fund Performnce: An Empiricl Decomposiion ino Sock-Picking len, Syle, rnscions Coss, nd Expenses, Journl of Finnce, 55, 1655 173. Wilson, R. B., 1968, On he heory of Syndices, Economeric, 36, 119 132. Wonder, N. X., 2, Conrcing on Rel-Opion Pyoffs, working pper, Universiy of Cliforni, Berkeley. 28